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INDICES [Exponents] - Questions & Answers

EXERCISE 7 (A)

1. Multiple Choice Type : Choose the correct answer from the options given below.

(a) (-2a² / b³)⁴ is equal to :
(i) -16a⁸/b¹² (ii) 16a⁸/b¹² (iii) 2a⁸/b¹² (iv) -2a⁸/b¹²

Apply the power of 4 to each term inside the bracket.
= (-2)⁴ × (a²)⁴ / (b³)⁴
= 16 × a^2×4 / b^3×4
= 16a⁸ / b¹²
Answer: (ii)

(b) ⁴√28 ÷ ⁴√7 is equal to :
(i) √2 (ii) √4 (iii) ⁴√2 (iv) ³√2

Combine the roots since they have the same index (4).
= ⁴√(28 / 7)
= ⁴√4
= (2²)^1/4
= 2^2/4 = 2^1/2
= √2
Answer: (i)

Express 25 and 9 as squares.
= ((5/3)²)^-3/2
Multiply the exponents.
= (5/3)^{2 × -3/2}
= (5/3)^-3
Invert the fraction to make the exponent positive.
= (3/5)³ = 27/125
Answer: (iii)

(d) 2 ÷ (243)^-1/5 is equal to :
(i) 2 ÷ 3 (ii) 1/6 (iii) 12 (iv) 6

Express 243 as 3⁵.
= 2 ÷ (3⁵)^-1/5
= 2 ÷ 3^-1
= 2 × 3¹
= 6
Answer: (iv)

(e) (0.01)^-1/2 :
(i) 10 (ii) 0.1 (iii) (0.1)^1/2 (iv) 0.0001

Convert the decimal to a power of 10.
= (10^-2)^-1/2
Multiply the exponents.
= 10^{-2 × -1/2}
= 10¹ = 10
Answer: (i)

(f) 3 × (32)^2/5 × 7⁰ is equal to :
(i) 0 (ii) 12 (iii) 1 (iv) 9

Any non-zero number to the power of 0 is 1 (7⁰ = 1).
Express 32 as 2⁵.
= 3 × (2⁵)^2/5 × 1
= 3 × 2² × 1
= 3 × 4
= 12
Answer: (ii)

2. Evaluate :
(i) 3³ × (243)^-2/3 × 9^-1/3

Express all bases as powers of 3.
= 3³ × (3⁵)^-2/3 × (3²)^-1/3
= 3³ × 3^-10/3 × 3^-2/3
Add the exponents since bases are the same.
= 3^{3 - 10/3 - 2/3}
= 3^{3 - 12/3}
= 3^{3 - 4}
= 3^-1 = 1/3

(ii) 5^-4 × (125)^5/3 ÷ (25)^-1/2

Express all bases as powers of 5.
= 5^-4 × (5³)^5/3 ÷ (5²)^-1/2
= 5^-4 × 5⁵ ÷ 5^-1
Apply product and quotient laws.
= 5^{-4 + 5 - (-1)}
= 5^{1 + 1}
= 5² = 25

(iii) (27/125)^2/3 × (9/25)^-3/2

Express terms as powers of (3/5).
= ((3/5)³)^2/3 × ((3/5)²)^-3/2
Multiply outer and inner exponents.
= (3/5)² × (3/5)^-3
Add the exponents.
= (3/5)^{2 - 3}
= (3/5)^-1 = 5/3

(iv) 7⁰ × (25)^-3/2 - 5^-3

Use 7⁰ = 1 and express 25 as 5².
= 1 × (5²)^-3/2 - 5^-3
= 5^-3 - 5^-3
Subtract identical terms.
= 0

(v) (16/81)^-3/4 × (49/9)^3/2 ÷ (343/216)^2/3

Convert bases to powers.
= ((2/3)⁴)^-3/4 × ((7/3)²)^3/2 ÷ ((7/6)³)^2/3
= (2/3)^-3 × (7/3)³ ÷ (7/6)²
Make negative exponents positive by reciprocating.
= (3/2)³ × (7/3)³ ÷ (7/6)²
= (27/8) × (343/27) × (36/49)
Cancel common factors.
= (343/8) × (36/49)
= 7 × (36/8)
= 7 × (9/2) = 63/2 = 31.5

3. Simplify :
(i) (8x³ ÷ 125y³)^2/3

Write the division as a fraction.
= ((2x)³ / (5y)³)^2/3
= ((2x / 5y)³)^2/3
Multiply the powers.
= (2x / 5y)²
= 4x² / 25y²

(ii) (a + b)^-1 ċ (a^-1 + b^-1)

Rewrite negative exponents as reciprocals.
= (1 / (a + b)) × (1/a + 1/b)
Find a common denominator for the second bracket.
= (1 / (a + b)) × ((b + a) / ab)
Cancel the common term (a + b).
= 1 / ab

(iii) (5ⁿ⁺³ - 6 × 5ⁿ⁺¹) / (9 × 5ⁿ - 5ⁿ × 2²)

Factor out the smallest power of 5 from the numerator (5ⁿ⁺¹) and denominator (5ⁿ).
= (5ⁿ⁺¹(5² - 6)) / (5ⁿ(9 - 4))
= (5ⁿ × 5¹ (25 - 6)) / (5ⁿ(5))
Cancel 5ⁿ.
= (5 × 19) / 5
Cancel 5.
= 19

(iv) (3x²)^-3 × (x⁹)^2/3

Distribute the exponents.
= 3^-3 × x^-6 × x⁶
= (1/27) × x^-6+6
= (1/27) × x⁰
= 1/27

4. Evaluate :
(i) √(1/4) + (0.01)^-1/2 - (27)^2/3

Simplify each term separately.
= 1/2 + (10^-2)^-1/2 - (3³)^2/3
= 0.5 + 10¹ - 3²
= 0.5 + 10 - 9
= 1.5 (or 3/2)

(ii) (27/8)^2/3 - (1/4)^-2 + 5⁰

Convert to power blocks.
= ((3/2)³)^2/3 - (2^-2)^-2 + 1
= (3/2)² - 2⁴ + 1
= 9/4 - 16 + 1
= 9/4 - 15
= 9/4 - 60/4 = -51/4

5. Simplify each of the following and express with positive index :
(i) (3^-4 / 2^-8)^1/4

Multiply the exponents of numerator and denominator by 1/4.
= 3^-1 / 2^-2
Invert positions to make exponents positive.
= 2² / 3¹ (or 4/3)

(ii) (27^-3 / 9^-3)^1/5

Combine terms with identical exponents first.
= ((27 / 9)^-3)^1/5
= (3^-3)^1/5
= 3^-3/5
Express with positive index.
= 1 / 3^3/5

(iii) (32)^-2/5 ÷ (125)^-2/3

Convert to prime bases.
= (2⁵)^-2/5 ÷ (5³)^-2/3
= 2^-2 ÷ 5^-2
Convert division to fraction.
= (1/2^{2}) ÷ (1/5^{2})
= 5² / 2² (or 25/4)

(iv) [1 - {1 - (1 - n)^-1}^-1]^-1

Start from the innermost bracket.
= [1 - {1 - 1/(1 - n)}^-1]^-1
Take LCM inside the curly bracket.
= [1 - {(1 - n - 1) / (1 - n)}^-1]^-1
= [1 - {-n / (1 - n)}^-1]^-1
Invert the fraction to remove the negative exponent.
= [1 - {(1 - n) / -n}]^-1
= [1 + (1 - n) / n]^-1
Take LCM again.
= [(n + 1 - n) / n]^-1
= [1 / n]^-1
= n

6. If 2160 = 2^a ċ 3^b ċ 5^c, find a, b and c. Hence calculate the value of 3^a × 2^-b × 5^-c. (HOTS)

Find the prime factorization of 2160.
2160 = 16 × 135 = 16 × 27 × 5
2160 = 2⁴ × 3³ × 5¹
By comparing powers, a = 4, b = 3, c = 1.
Substitute these into the given expression:
= 3⁴ × 2^-3 × 5^-1
= 81 × (1/8) × (1/5)
= 81 / 40

7. If 1960 = 2^a ċ 5^b ċ 7^c, calculate the value of 2^-a ċ 7^b ċ 5^-c.

Find prime factorization of 1960.
1960 = 8 × 245 = 8 × 5 × 49
1960 = 2³ × 5¹ × 7²
By comparing powers, a = 3, b = 1, c = 2.
Substitute into expression:
= 2^-3 × 7¹ × 5^-2
= 7 / (2³ × 5²)
= 7 / (8 × 25) = 7 / 200

8. Simplify :
(i) (8^3a × 2⁵ × 2^2a) / (4 × 2^11a × 2^-2a)

Convert 8 and 4 to bases of 2.
= ((2³)^3a × 2^5+2a) / (2² × 2^{11a - 2a})
= (2^9a × 2^5+2a) / (2² × 2^9a)
Add the exponents in numerator.
= 2^11a+5 / 2^9a+2
Subtract denominator exponents from numerator exponents.
= 2^{(11a+5) - (9a+2)}
= 2^2a+3

(ii) (3 × 27ⁿ⁺¹ + 9 × 3^3n-1) / (8 × 3³ⁿ - 5 × 27ⁿ)

Convert 27 and 9 to bases of 3.
= (3¹ × (3³)ⁿ⁺¹ + 3² × 3^3n-1) / (8 × 3³ⁿ - 5 × (3³)ⁿ)
= (3¹ × 3³ⁿ⁺³ + 3³ⁿ⁺¹) / (8 × 3³ⁿ - 5 × 3³ⁿ)
= (3³ⁿ⁺⁴ + 3³ⁿ⁺¹) / (3³ⁿ(8 - 5))
Factor out 3³ⁿ⁺¹ from numerator.
= (3³ⁿ⁺¹(3³ + 1)) / (3³ⁿ × 3¹)
= (3³ⁿ⁺¹(27 + 1)) / (3³ⁿ⁺¹)
Cancel common terms.
= 28

9. Show that :
(a^m / a^-n)^m-n × (aⁿ / a^-l)^n-l × (a^l / a^-m)^l-m = 1

L.H.S: Bring negative exponents up to add them.
= (a^m+n)^m-n × (a^n+l)^n-l × (a^l+m)^l-m
Use identity (x+y)(x-y) = x² - y².
= a^m²-n² × a^n²-l² × a^l²-m²
Add all exponents as bases are same.
= a^{m² - n² + n² - l² + l² - m²}
= a⁰ = 1 = R.H.S.
Hence Proved.

10. If a = x^m+n · y^l ; b = x^n+l · y^m and c = x^l+m · yⁿ, prove that : a^m-n · b^n-l · c^l-m = 1

L.H.S = (x^m+ny^l)^m-n · (x^n+ly^m)^n-l · (x^l+myⁿ)^l-m
Distribute the powers.
= x^(m+n)(m-n)y^l(m-n) · x^(n+l)(n-l)y^m(n-l) · x^(l+m)(l-m)y^n(l-m)
= x^m²-n²y^lm-ln · x^n²-l²y^mn-ml · x^l²-m²y^nl-nm
Combine all x bases and y bases.
= x^{m²-n²+n²-l²+l²-m²} · y^{lm-ln+mn-ml+nl-nm}
All terms in the exponents cancel out.
= x⁰ · y⁰ = 1 × 1 = 1 = R.H.S.
Hence Proved.

11. Simplify :
(i) (x^a / x^b)^a²+ab+b² × (x^b / x^c)^b²+bc+c² × (x^c / x^a)^c²+ca+a²

Use quotient law inside brackets.
= (x^a-b)^a²+ab+b² × (x^b-c)^b²+bc+c² × (x^c-a)^c²+ca+a²
Apply the formula (u-v)(u²+uv+v²) = u³-v³.
= x^a³-b³ × x^b³-c³ × x^c³-a³
Add the exponents.
= x^{a³ - b³ + b³ - c³ + c³ - a³}
= x⁰ = 1

(ii) (x^a / x^-b)^a²-ab+b² × (x^b / x^-c)^b²-bc+c² × (x^c / x^-a)^c²-ca+a²

Bring up negative exponents to change signs.
= (x^a+b)^a²-ab+b² × (x^b+c)^b²-bc+c² × (x^c+a)^c²-ca+a²
Apply the formula (u+v)(u²-uv+v²) = u³+v³.
= x^a³+b³ × x^b³+c³ × x^c³+a³
Add the exponents.
= x^{a³ + b³ + b³ + c³ + c³ + a³}
= x^{2(a³+b³+c³)}

EXERCISE 7 (B)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) a^-1 / (a^-1 + b^-1) is equal to :
(i) b / (a+b) (ii) (a+b) / a (iii) a / (a+b) (iv) a(a+b)

Convert negative exponents to fractions.
= (1/a) / (1/a + 1/b)
= (1/a) / ((b+a) / ab)
= (1/a) × (ab / (a+b))
= b / (a+b)
Answer: (i)

(b) If 4^2x = 1/32, the value of x is :
(i) 1.25 (ii) -1.25 (iii) 1 (iv) -1

Convert bases to 2.
(2²)^2x = 2^-5
2^4x = 2^-5
Equate exponents: 4x = -5
x = -5/4 = -1.25
Answer: (ii)

Move negative exponents from denominator to numerator.
= 3x(y¹) + 2y(x¹)
= 3xy + 2xy
= 5xy
Answer: (iii)

(d) (8^-4/3 ÷ 2^-2)^1/2 is equal to :
(i) 1/4 (ii) -1/4 (iii) -1/2 (iv) 1/2

Express 8 as 2³.
= ((2³)^-4/3 ÷ 2^-2)^1/2
= (2^-4 ÷ 2^-2)^1/2
= (2^{-4 - (-2)})^1/2
= (2^-2)^1/2 = 2^-1 = 1/2
Answer: (iv)

(e) If 8^2x+5 = 1, value of x is :
(i) -5/2 (ii) 5/2 (iii) 2 (iv) 2/5

Since a⁰ = 1, we can write 1 as 8⁰.
8^2x+5 = 8⁰
2x + 5 = 0
2x = -5 ⇒ x = -5/2
Answer: (i)

(f) If 3^x+1 = 9^x-2, the value of x is :
(i) -5 (ii) 5 (iii) 0 (iv) 3

Express 9 as 3².
3^x+1 = (3²)^x-2
3^x+1 = 3^2x-4
Equating exponents: x + 1 = 2x - 4
x = 5
Answer: (ii)

(g) If √(a/b) = (b/a)^1-2x, the value of x is :
(i) 3/4 (ii) 3/5 (iii) -3/4 (iv) -3/5

Convert root to power and flip the right base.
(a/b)^1/2 = (a/b)^-(1-2x)
Equate exponents: 1/2 = 2x - 1
2x = 3/2 ⇒ x = 3/4
Answer: (i)

2. Solve for x :
(i) 2^2x+1 = 8

2^2x+1 = 2³
2x + 1 = 3
2x = 2
x = 1

(ii) 2^5x-1 = 4 × 2^3x+1

2^5x-1 = 2² × 2^3x+1
2^5x-1 = 2^3x+3
5x - 1 = 3x + 3
2x = 4
x = 2

(iii) 3^4x+1 = (27)^x+1

3^4x+1 = (3³)^x+1
3^4x+1 = 3^3x+3
4x + 1 = 3x + 3
x = 2

(iv) (49)^x+4 = 7² × (343)^x+1

(7²)^x+4 = 7² × (7³)^x+1
7^2x+8 = 7² × 7^3x+3
7^2x+8 = 7^3x+5
2x + 8 = 3x + 5
x = 3

3. Find x, if :
(i) 4^2x = 1/32

(2²)^2x = 2^-5
2^4x = 2^-5
4x = -5
x = -5/4

(ii) √(2^x+3) = 16

2^(x+3)/2 = 2⁴
(x+3) / 2 = 4
x + 3 = 8
x = 5

(iii) (√(3/5))^x+1 = 125/27

(3/5)^(x+1)/2 = (5/3)³
(3/5)^(x+1)/2 = (3/5)^-3
(x+1) / 2 = -3
x + 1 = -6
x = -7

(iv) (³√(2/3))^x-1 = 27/8

(2/3)^(x-1)/3 = (3/2)³
(2/3)^(x-1)/3 = (2/3)^-3
(x-1) / 3 = -3
x - 1 = -9
x = -8

4. Solve :
(i) 4^x-2 - 2^x+1 = 0

(2²)^x-2 = 2^x+1
2^2x-4 = 2^x+1
2x - 4 = x + 1
x = 5

(ii) 3^x² : 3^x = 9 : 1

3^x² / 3^x = 9
3^x²-x = 3²
x² - x = 2
x² - x - 2 = 0
(x - 2)(x + 1) = 0
x = 2, -1

5. Solve :
(i) 8 × 2^2x + 4 × 2^x+1 = 1 + 2^x

Let 2^x = y.
8y² + 4 × 2y - y - 1 = 0
8y² + 8y - y - 1 = 0
8y(y + 1) - 1(y + 1) = 0
(8y - 1)(y + 1) = 0
y = 1/8 or y = -1 (reject negative for real x).
2^x = 2^-3
x = -3

(ii) 2^2x + 2^x+2 - 4 × 2³ = 0

Let 2^x = y.
y² + 4y - 32 = 0
(y + 8)(y - 4) = 0
y = 4 (reject y = -8).
2^x = 2²
x = 2

(iii) (√3)^x-3 = (⁴√3)^x+1

3^(x-3)/2 = 3^(x+1)/4
(x - 3) / 2 = (x + 1) / 4
Multiply by 4.
2(x - 3) = x + 1
2x - 6 = x + 1
x = 7

6. Find the values of m and n if :
4^2m = (³√16)^-6/n = (√8)²

Simplify rightmost term: (√8)² = 8 = 2³.
Solve for m: 4^2m = 2³ ⇒ (2²)^2m = 2³ ⇒ 2^4m = 2³ ⇒ 4m = 3 ⇒ m = 3/4
Solve for n: (16^1/3)^-6/n = 2³ ⇒ 16^-2/n = 2³
(2⁴)^-2/n = 2³ ⇒ 2^-8/n = 2³
-8/n = 3 ⇒ n = -8/3

7. Solve for x and y, if :
(√32)^x ÷ 2^y+1 = 1 and 8^y - 16^4-x/2 = 0

Equation 1: (2^5/2)^x = 2^y+1 ⇒ 5x/2 = y + 1 ⇒ 5x - 2y = 2.
Equation 2: (2³)^y = (2⁴)^4-x/2 ⇒ 3y = 16 - 2x ⇒ 2x + 3y = 16.
Solve simultaneously: Multiply Eq1 by 3 and Eq2 by 2.
15x - 6y = 6
4x + 6y = 32
Add them: 19x = 38 ⇒ x = 2.
Substitute x in Eq1: 5(2) - 2y = 2 ⇒ 10 - 2 = 2y ⇒ 2y = 8 ⇒ y = 4.

8. Prove that :
(i) (x^a / x^b)^a+b-c (x^b / x^c)^b+c-a (x^c / x^a)^c+a-b = 1

= x^(a-b)(a+b-c) · x^(b-c)(b+c-a) · x^(c-a)(c+a-b)
= x^{a²-b²-ac+bc} · x^{b²-c²-ab+ac} · x^{c²-a²-bc+ab}
Add the exponents.
= x^{a²-b²-ac+bc+b²-c²-ab+ac+c²-a²-bc+ab}
= x⁰ = 1
Hence Proved.

(ii) x^a(b-c) / x^b(a-c) ÷ (x^b / x^a)^c = 1

= (x^ab-ac / x^ab-bc) ÷ (x^bc / x^ac)
= x^{ab-ac - (ab-bc)} ÷ x^bc-ac
= x^bc-ac ÷ x^bc-ac
= 1
Hence Proved.

9. If a^x = b, b^y = c and c^z = a, prove that : xyz = 1.

Start with the last equation: a = c^z.
Substitute c with b^y: a = (b^y)^z = b^yz.
Substitute b with a^x: a = (a^x)^yz = a^xyz.
Since a¹ = a^xyz, comparing the exponents yields:
xyz = 1. (Hence Proved)

10. If a^x = b^y = c^z and b² = ac, prove that : y = 2xz / (x+z) (HOTS)

Let a^x = b^y = c^z = k.
Then a = k^1/x, b = k^1/y, c = k^1/z.
Given b² = ac, substitute a, b, c.
(k^1/y)² = (k^1/x) · (k^1/z)
k^2/y = k^{1/x + 1/z}
Equate exponents: 2/y = 1/x + 1/z
2/y = (z + x) / xz
Invert and isolate y: y/2 = xz / (x + z)
y = 2xz / (x + z). (Hence Proved)

11. If 5^-p = 4^-q = 20^r; show that : 1/p + 1/q + 1/r = 0. (BM)

Let 5^-p = 4^-q = 20^r = k.
Then 5 = k^-1/p, 4 = k^-1/q, and 20 = k^1/r.
We know that 5 × 4 = 20.
Substitute the k values: k^-1/p × k^-1/q = k^1/r
k^{-(1/p + 1/q)} = k^1/r
Equate exponents: -(1/p + 1/q) = 1/r
1/p + 1/q + 1/r = 0. (Hence Proved)

12. If m ≠ n and (m + n)^-1(m^-1 + n^-1) = m^xn^y; show that : x + y + 2 = 0.

L.H.S = (1 / (m + n)) × (1/m + 1/n)
= (1 / (m + n)) × ((n + m) / mn)
Cancel (m+n): = 1 / mn = m^-1n^-1
So, m^-1n^-1 = m^xn^y
Comparing exponents: x = -1, y = -1.
Therefore, x + y + 2 = -1 - 1 + 2 = 0. (Hence Proved)

13. If 5^x+1 = 25^x-2; find the value of : 3^x-3 × 2^3-x.

Find x: 5^x+1 = (5²)^x-2 ⇒ 5^x+1 = 5^2x-4.
x + 1 = 2x - 4 ⇒ x = 5.
Substitute x in expression: 3^5-3 × 2^3-5
= 3² × 2^-2
= 9 × (1/4) = 9/4

14. If 4^x+3 = 112 + 8 × 4^x; find (18x)^3x.

4^x · 4³ - 8 × 4^x = 112
64 · 4^x - 8 · 4^x = 112
56 · 4^x = 112
4^x = 2 ⇒ 2^2x = 2¹ ⇒ 2x = 1 ⇒ x = 1/2.
Substitute x: (18 × 1/2)^3(1/2)
= 9^3/2 = (3²)^3/2 = 3³ = 27

15. Solve for x :
(i) 4^x-1 × (0.5)^3-2x = (1/8)^-x

Convert to bases of 2.
(2²)^x-1 × (2^-1)^3-2x = (2^-3)^-x
2^2x-2 × 2^-3+2x = 2^3x
2^4x-5 = 2^3x
4x - 5 = 3x ⇒ x = 5

(ii) (a^3x+5)² · (a^x)⁴ = a^8x+12

a^6x+10 · a^4x = a^8x+12
a^10x+10 = a^8x+12
10x + 10 = 8x + 12
2x = 2 ⇒ x = 1

(iii) (81)^3/4 - (1/32)^-2/5 + x(1/2)^-1 · 2⁰ = 27

(3⁴)^3/4 - (2^-5)^-2/5 + x(2¹)(1) = 27
3³ - 2² + 2x = 27
27 - 4 + 2x = 27
23 + 2x = 27
2x = 4 ⇒ x = 2

(iv) 2^3x+3 = 2^3x+1 + 48

2^3x · 2³ - 2^3x · 2¹ = 48
2^3x(8 - 2) = 48
2^3x(6) = 48
2^3x = 8 = 2³
3x = 3 ⇒ x = 1

(v) 3(2^x + 1) - 2^x+2 + 5 = 0

3 · 2^x + 3 - 4 · 2^x + 5 = 0
-1 · 2^x + 8 = 0
2^x = 8 = 2³
x = 3

(vi) 9^x+2 = 720 + 9^x

9^x · 9² - 9^x = 720
9^x(81 - 1) = 720
9^x(80) = 720
9^x = 9 = 9¹
x = 1

TEST YOURSELF

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) (20⁰ - 18⁰) × 7⁰ is equal to :
(i) 0 (ii) 1 (iii) 14 (iv) none of these

= (1 - 1) × 1
= 0 × 1 = 0
Answer: (i)

(b) (-2)^-1 ÷ (-2)^-4 is equal to :
(i) 8 (ii) 1/8 (iii) -8 (iv) -1/8

= (-2)^{-1 - (-4)}
= (-2)³ = -8
Answer: (iii)

Find a/b: = 2/3 ÷ 1 = 2/3.
Evaluate (a/b)^-2: = (2/3)^-2
= (3/2)² = 9/4
Answer: (iii)

(d) If 5^2x+3 = 1, the value of x is :
(i) 3/2 (ii) -3/2 (iii) 2/3 (iv) -2/3

5^2x+3 = 5⁰
2x + 3 = 0 ⇒ x = -3/2
Answer: (ii)

(e) (2 + 3)^-1 × (2^-1 + 3^-1) is equal to :
(i) 6 (ii) -6 (iii) 1/6 (iv) -1/6

= 5^-1 × (1/2 + 1/3)
= (1/5) × (5/6)
= 1/6
Answer: (iii)

(f) Statement (1) : (3/4)^-4 × (3/4)^-5 = (3/4)^3x ⇒ x = -1
Statement (2) : (3/4)^-4-5 = (3/4)^3x ⇒ 3x = -9
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true.

Check 1: L.H.S = (3/4)^-9. So 3x = -9 ⇒ x = -3. Statement 1 incorrectly claims x = -1. False.
Check 2: L.H.S = (3/4)^-9. Equating powers gives 3x = -9. True.
Answer: (iv)

(g) Statement (1) : (5/8)^-7 × (8/5)^-4 = x ⇒ x = (5/8)³
Statement (2) : (5/8)^-7 × (8/5)^-4 = x ⇒ x = (8/5)³
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true.

Simplify: = (5/8)^-7 × (5/8)⁴ = (5/8)^-3 = (8/5)³.
So Statement 1 is false and Statement 2 is true.
Answer: (iv)

(h) Assertion (A) : (3^-7 ÷ 3^-10) × 3^-5 = 1/9 .
Reason (R) : 1/3⁷ × 3¹⁰ × 1/3⁵ .
(i) A is true, R is false.
(ii) A is false, R is true.
(iii) Both A and R are true and R is the correct reason for A.
(iv) Both A and R are true and R is the incorrect reason for A.

Evaluate A: = (3³) × 3^-5 = 3^-2 = 1/9. True.
Evaluate R: This is the expanded form which also equals 1/9. True.
R is a valid explanatory step for A.
Answer: (iii)

(i) Assertion (A) : (1³ + 2³ + 3³)^1/2 = x, then x = √1 + √8 + √27.
Reason (R) : x = (1 + 8 + 27)^1/2 = (36)^1/2 = 6
(i) A is true, R is false.
(ii) A is false, R is true.
(iii) Both A and R are true and R is the correct reason for A.
(iv) Both A and R are true and R is the incorrect reason for A.

Check A: (1 + 8 + 27)^1/2 = (36)^1/2 = 6. But √1 + √8 + √27 ≠ 6. False.
Check R: x = 6 is perfectly calculated. True.
Answer: (ii)

2. Evaluate :
(i) 9^5/2 - 3 × 8⁰ - (1/81)^-1/2

= (3²)^5/2 - 3(1) - (81)^1/2
= 3⁵ - 3 - 9
= 243 - 12 = 231

(ii) (64)^2/3 - ³√125 - 1 / 2^-5 + (27)^-2/3 × (25/9)^-1/2

= (4³)^2/3 - 5 - 2⁵ + (3³)^-2/3 × (9/25)^1/2
= 4² - 5 - 32 + 3^-2 × (3/5)
= 16 - 5 - 32 + (1/9)(3/5)
= -21 + 1/15
= (-315 + 1)/15 = -314/15

(iii) [(-2/3)^-2]³ × (1/3)^-4 × 3^-1 × 1/6

= (-2/3)^-6 × 3⁴ × 1/3 × 1/6
= (-3/2)⁶ × 3³ × 1/6
= (729/64) × 27 × (1/6)
= (3⁶/2⁶) × 3³ × (1 / (2 · 3))
= 3⁸ / 2⁷ = 6561 / 128

3. Simplify : (3 × 9ⁿ⁺¹ - 9 × 3²ⁿ) / (3 × 3²ⁿ⁺³ - 9ⁿ⁺¹)

Convert all to bases of 3.
= (3 × (3²)ⁿ⁺¹ - 3² × 3²ⁿ) / (3²ⁿ⁺⁴ - (3²)ⁿ⁺¹)
= (3²ⁿ⁺³ - 3²ⁿ⁺²) / (3²ⁿ⁺⁴ - 3²ⁿ⁺²)
Factor out 3²ⁿ⁺².
= (3²ⁿ⁺²(3 - 1)) / (3²ⁿ⁺²(3² - 1))
= 2 / (9 - 1) = 2 / 8 = 1/4

4. Solve : 3^x-1 × 5^2y-3 = 225.

Express 225 as prime factors: 225 = 9 × 25 = 3² × 5².
3^x-1 × 5^2y-3 = 3² × 5².
Equating base 3 powers: x - 1 = 2 ⇒ x = 3.
Equating base 5 powers: 2y - 3 = 2 ⇒ 2y = 5 ⇒ y = 5/2.

5. If (a^-1b² / a²b^-4)⁷ ÷ (a³b^-5 / a^-2b³)^-5 = a^x · b^y, find x + y.

Simplify inside first bracket: (a^-3b⁶)⁷ = a^-21b⁴².
Simplify inside second bracket: (a⁵b^-8)^-5 = a^-25b⁴⁰.
Divide them: a^-21b⁴² ÷ a^-25b⁴⁰ = a^{-21 - (-25)}b^{42 - 40} = a⁴b².
Comparing with a^xb^y gives x = 4, y = 2.
x + y = 4 + 2 = 6.

6. If 3^x+1 = 9^x-3, find the value of 2^1+x.

3^x+1 = (3²)^x-3 = 3^2x-6
x + 1 = 2x - 6 ⇒ x = 7.
Substitute x: 2¹⁺⁷ = 2⁸ = 256.

7. If 2^x = 4^y = 8^z and 1/2x + 1/4y + 1/8z = 4, find the value of x.

2^x = 2^2y = 2^3z ⇒ x = 2y = 3z.
Therefore, y = x/2 and z = x/3.
Substitute in the equation: 1/2x + 1/(4(x/2)) + 1/(8(x/3)) = 4
1/2x + 1/2x + 3/8x = 4
Multiply by 8x: 4 + 4 + 3 = 32x
11 = 32x ⇒ x = 11/32

8. If (9ⁿ · 3² · 3ⁿ - (27)ⁿ) / (3^m · 2)³ = 3^-3. Show that : m - n = 1.

Simplify numerator: 3²ⁿ · 3² · 3ⁿ - 3³ⁿ = 3³ⁿ⁺² - 3³ⁿ = 3³ⁿ(3² - 1) = 3³ⁿ × 8.
Simplify denominator: 3^3m · 2³ = 3^3m × 8.
Fraction becomes: (3³ⁿ × 8) / (3^3m × 8) = 3^3n-3m.
So, 3^3(n-m) = 3^-3
3(n-m) = -3 ⇒ n - m = -1 ⇒ m - n = 1. (Hence Proved)

9. Solve for x : (13)^√x = 4⁴ - 3⁴ - 6.

Calculate right side: 256 - 81 - 6 = 169.
13^√x = 169 = 13².
√x = 2 ⇒ x = 4.

10. If 3^4x = (81)^-1 and (10)^1/y = 0.0001, find the value of 2^-x × 16^y.

Solve for x: 3^4x = 3^-4 ⇒ 4x = -4 ⇒ x = -1.
Solve for y: 10^1/y = 10^-4 ⇒ 1/y = -4 ⇒ y = -1/4.
Substitute: 2^-(-1) × 16^-1/4 = 2¹ × (2⁴)^-1/4
= 2 × 2^-1 = 2 × (1/2) = 1.

11. If (a^m)ⁿ = a^m · aⁿ, find the value of : m(n - 1) - (n - 1)

From the given equation: a^mn = a^m+n.
So, mn = m + n, which means mn - m - n = 0.
Expand expression: mn - m - n + 1.
Substitute 0: 0 + 1 = 1.

12. If m = ³√15 and n = ³√14, find the value of m - n - 1 / (m² + mn + n²) (HOTS)

Notice m³ = 15 and n³ = 14.
We know m³ - n³ = (m - n)(m² + mn + n²).
Substitute values: 15 - 14 = 1 = (m - n)(m² + mn + n²).
So, m - n = 1 / (m² + mn + n²).
The expression becomes: (m - n) - (m - n) = 0.

13. Evaluate : (x^q / x^r)^1/qr × (x^r / x^p)^1/rp × (x^p / x^q)^1/pq

= (x^q-r)^1/qr × (x^r-p)^1/rp × (x^p-q)^1/pq
= x^(q-r)/qr × x^(r-p)/rp × x^(p-q)/pq
= x^{(1/r - 1/q)} × x^{(1/p - 1/r)} × x^{(1/q - 1/p)}
Add the exponents.
= x⁰ = 1

14. Prove that :
(i) a^-1 / (a^-1 + b^-1) + a^-1 / (a^-1 - b^-1) = 2b² / (b² - a²)

Change negative exponents to fractions.
= (1/a) / (1/a + 1/b) + (1/a) / (1/a - 1/b)
= (1/a) / ((b+a)/ab) + (1/a) / ((b-a)/ab)
= b / (b+a) + b / (b-a)
Take common denominator.
= b((b-a) + (b+a)) / (b² - a²)
= b(2b) / (b² - a²) = 2b² / (b² - a²).
Hence Proved.

(ii) (a+b+c) / (a^-1b^-1 + b^-1c^-1 + c^-1a^-1) = abc

Denominator = 1/ab + 1/bc + 1/ca.
Take LCM of denominator = (c + a + b) / abc.
Expression = (a + b + c) / ((a + b + c) / abc).
The (a+b+c) term cancels out.
= abc.
Hence Proved.

15. Find the value of x : (3+4)(3²+4²)(3⁴+4⁴)(3⁸+4⁸)(3¹⁶+4¹⁶)(3³²+4³²) = (4^x - 3^x)

Multiply L.H.S by (4 - 3), which equals 1, so it doesn't change the value.
= (4 - 3)(4 + 3)(4² + 3²)...(4³² + 3³²)
Use (a-b)(a+b) = a²-b² repeatedly.
= (4² - 3²)(4² + 3²)...
= (4⁴ - 3⁴)...
This cascades until the last term.
= (4⁶⁴ - 3⁶⁴)
Compare with (4^x - 3^x).
x = 64

Case-Study Based Question
Mr. Mohan divided a sum of money into three parts 5^x, 3^y and 2^z and distributed among his three sons Aman, Aryan and Asmit respectively. The product of their share is 12,96,000.

Prime factorize the product 12,96,000.
12,96,000 = 1296 × 1000 = 6⁴ × 10³
= (2 × 3)⁴ × (2 × 5)³
= 2⁴ × 3⁴ × 2³ × 5³ = 2⁷ × 3⁴ × 5³.
Comparing with 5^x · 3^y · 2^z gives x = 3, y = 4, z = 7.

(i) Find the share of Aman.

Aman's share = 5^x = 5³ = 125.

(ii) Find the sum of shares of Aryan and Asmit.

Aryan = 3⁴ = 81. Asmit = 2⁷ = 128.
Sum = 81 + 128 = 209.

(iii) Find the value of (x + y + z).

= 3 + 4 + 7 = 14.

(iv) Find the value of (√x^y)^z-5.

Substitute x, y, z.
= (√3⁴)^7-5
= (3²)² = 9² = 81.

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Quick Review Flashcards - Click to flip and test your knowledge!

Question

In the expression $a^m$, what is the term $a$ called?

Answer

The base.

Question

In the expression $a^m$, what are the three alternative names for the term $m$?

Answer

Power, exponent, or index.

Question

How is the mathematical expression $a^m$ read aloud?

Answer

It is read as '$a$ power $m$' or '$a$ raised to the power $m$'.

Question

Write the product $a \times a \times a \dots$ to $m$ terms using index notation.

Answer

$a^m$

Question

State the 1st Law of Indices (Product Law).

Answer

$a^m \times a^n = a^{m+n}$

Question

State the 2nd Law of Indices (Quotient Law) for the expression $\frac{a^m}{a^n}$.

Answer

$\frac{a^m}{a^n} = a^{m-n}$

Question

State the 3rd Law of Indices (Power Law) for the expression $(a^m)^n$.

Answer

$(a^m)^n = a^{mn}$

Question

Simplify the expression $a^7 \times a^4$.

Answer

$a^{11}$

Question

Simplify the expression $a^3 \times a^{-6}$.

Answer

$a^{-3}$

Question

Simplify the expression $\frac{a^7}{a^4}$.

Answer

$a^3$

Question

Simplify the expression $\frac{a^3}{a^6}$.

Answer

$a^{-3}$

Question

Simplify the expression $(a^3)^4$.

Answer

$a^{12}$

Question

According to the laws for handling indices, what does $(a \times b)^m$ expand to?

Answer

$a^m \times b^m$

Question

According to the laws for handling indices, what does $(\frac{a}{b})^m$ expand to?

Answer

$\frac{a^m}{b^m}$

Question

If $a \ne 0$ and $n$ is a positive integer, express $\sqrt[n]{a}$ in index form.

Answer

$a^{1/n}$

Question

Express the square root $\sqrt{a}$ in index form.

Answer

$a^{1/2}$

Question

Express the cube root $\sqrt[3]{a}$ in index form.

Answer

$a^{1/3}$

Question

Convert the fractional index expression $a^{m/n}$ into radical form.

Answer

$\sqrt[n]{a^m}$

Question

Express $a^{4/5}$ in radical form.

Answer

$\sqrt[5]{a^4}$

Question

Express $5^{2/3}$ in radical form.

Answer

$\sqrt[3]{5^2}$

Question

For any non-zero number $a$, what is the equivalent of $a^{-n}$ with a positive index?

Answer

$\frac{1}{a^n}$

Question

For any non-zero number $a$, what is the equivalent of $a^n$ using a negative index?

Answer

$\frac{1}{a^{-n}}$

Question

What is the value of any non-zero number raised to the power zero ($a^0$)?

Answer

1 (or unity).

Question

Rule: If $m$ is an even number, what is the result of $(-a)^m$?

Answer

$a^m$

Question

Rule: If $m$ is an odd number, what is the result of $(-a)^m$?

Answer

$-a^m$

Question

Evaluate the numerical value of $(-2)^4$.

Answer

$16$ (or $2^4$)

Question

Evaluate the numerical value of $(-2)^5$.

Answer

$-32$ (or $-2^5$)

Question

Evaluate $27^{-1/3}$.

Answer

$\frac{1}{3}$

Question

Evaluate $9^{3/2}$.

Answer

$27$

Question

Simplify $(\frac{1}{81})^{-1/2}$.

Answer

$9$ (or $81^{1/2}$)

Question

In the process of solving exponential equations, if $a^x = a^y$, what can be concluded about $x$ and $y$?

Answer

$x = y$

Question

When simplifying an expression like $\frac{3^{a+2} - 3^{a+1}}{4 \times 3^a - 3^a}$, what is the first step to isolate $3^a$?

Answer

Factor out $3^a$ from the numerator and the denominator.

Question

Simplify the expression $(a^{m-n})^{m+n} \cdot (a^{n-l})^{n+l} \cdot (a^{l-m})^{l+m}$.

Answer

Question

To solve the equation $9 \times 3^x = (27)^{2x-5}$, into what common base should both sides be converted?

Answer

Base 3.

Question

When solving $2^{2x+3} - 9 \times 2^x + 1 = 0$, what substitution is recommended to simplify the equation into a quadratic?

Answer

Substitute $2^x = y$.

Question

Evaluate the value of $(2 \times 3)^5$.

Answer

$2^5 \times 3^5$

Question

Convert $\sqrt[3]{a^5}$ into index form.

Answer

$a^{5/3}$

Question

Convert $\sqrt{10}$ into index form.

Answer

$10^{1/2}$

Question

If $1176 = 2^p \cdot 3^q \cdot 7^r$, what are the values of $p, q$, and $r$ after prime factorisation?

Answer

$p = 3, q = 1, r = 2$

Question

Evaluate $32^{0.8}$ by converting the decimal index to a fraction.

Answer

$16$

Question

What is the value of $(0.8)^{-1}$ as a fraction?

Answer

$\frac{5}{4}$ (or $1.25$)

Question

Simplify the quotient $(\frac{2}{3})^{-2}$.

Answer

$\frac{9}{4}$

Question

Simplify $a^n \cdot a^{-n}$.

Answer

Question

How is the expression $(\sqrt{x})^y$ represented in index form?

Answer

$x^{y/2}$

Question

In an exponential equation, if the bases on both sides are equal, what must be done with the exponents?

Answer

The exponents must be equated.

Question

Solve for $x$: $2^{2x+1} = 8$.

Answer

$x = 1$

Question

Solve for $x$: $4^{2x} = \frac{1}{32}$.

Answer

$x = -1.25$ (or $-\frac{5}{4}$)

Question

What does the expression $x^{a-b} \cdot x^{b-c} \cdot x^{c-a}$ simplify to?

Answer

Question

Simplify $(\frac{2a^2}{b^3})^4$.

Answer

$\frac{16a^8}{b^{12}}$

Question

What is the value of $(20^0 - 18^0) \times 7^0$?

Answer

Question

Simplify $(a+b)^{-1} \cdot (a^{-1} + b^{-1})$.

Answer

$\frac{1}{ab}$

Question

If $x^a = y, y^b = z$, and $z^c = x$, what is the value of $abc$?

Answer

Question

Simplify $(\frac{x^a}{x^b})^{a+b} \cdot (\frac{x^b}{x^c})^{b+c} \cdot (\frac{x^c}{x^a})^{c+a}$.

Answer

Question

Determine the value of $x$ if $8^{2x+5} = 1$.

Answer

$x = -2.5$ (or $-\frac{5}{2}$)

Question

Simplify $2^3 \times (243)^{-2/3} \times 9^{-1/3}$ by converting to base 3.

Answer

$\frac{8}{27}$

Question

Convert the radical expression $\sqrt[n]{x^m}$ into index form.

Answer

$x^{m/n}$

Question

Evaluate $5^{-4} \times (125)^{5/3} \div (25)^{-1/2}$.

Answer

Question

Simplify $(3x^2)^{-3} \times (x^9)^{2/3}$.

Answer

$\frac{1}{27}$

Question

If $a^x = b^y = c^z$ and $b^2 = ac$, what is the relationship between $x, y$, and $z$?

Answer

$\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$

Question

Evaluate $(0.01)^{1/2}$.

Answer

0.1