Class 8 Maths Solution | Lesson 9 Indices | Curriculum Development Centre (CDC)

Lesson 9 Indices

Exercise 9

1. Simplify using the law of indices.

(a) 3⁴ × 3³

= 3⁽⁴⁺³⁾

= 3⁷

(b) x³ × x⁵

= x⁽³⁺⁵⁾

= x⁸

(c) ab⁴ × b³

= a × b⁽⁴⁺³⁾

= ab⁷

(d) (a²b) × (ab³)

= a² × a × b × b³

= a³b⁴

(e) 3x⁴ × 2x³

= (3 × 2) × (x⁴ × x³)

= 6x⁽⁴⁺³⁾

= 6x⁷

(f) (–2x⁴) × (3x³)

= (–2 × 3) × (x⁴ × x³)

= –6x⁽⁴⁺³⁾

= –6x⁷

(g) (ab) × (a³b³) × (a²b)

= (a × a³ × a²) × (b × b³ × b)

= a⁽¹⁺³⁺²⁾ × b⁽¹⁺³⁺¹⁾

= a⁶b⁵

2. Simplify using the law of indices.

(a) 4⁴ ÷ 4²

= 4⁽⁴⁻²⁾

= 4²

(b) x⁸ ÷ x⁵

= x⁽⁸⁻⁵⁾

= x³

(c) a⁴b⁴ ÷ a³b³

= a⁽⁴⁻³⁾ × b⁽⁴⁻³⁾

= ab

(d) (x⁶y³) ÷ (x³y³)

= x⁽⁶⁻³⁾ × y⁽³⁻³⁾

= x³

(e) 8x⁴ ÷ 2x³

= (8 ÷ 2) × (x⁴ ÷ x³)

= 4x⁽⁴⁻³⁾

= 4x

(f) 16x⁴ ÷ 8x³

= (16 ÷ 8) × (x⁴ ÷ x³)

= 2x⁽⁴⁻³⁾

= 2x

3. Simplify using the law of indices.

(a) (3a)⁰

= 1 (Any non-zero number raised to the power of 0 is 1)

(b) (2b)³

= 2³ × b³

= 8b³

(c) (–3x)⁴

= (–3)⁴ × x⁴

= 81x⁴

(d) (–4ab²)³

= (–4)³ × a³ × (b²)³

= –64a³b⁶

(e) (3a³b²)²

= 3² × (a³)² × (b²)²

= 9a⁶b⁴

(f) (x²y²)²

= (x²)² × (y²)²

= x⁴y⁴

(g) (3xy)² ÷ 3xy

= 3² × x² × y² ÷ 3xy

= 3 × x × y

= 3xy

(h) a⁴ⁿ⁻² ÷ a²⁽²ⁿ⁻¹⁾

= a⁽⁴ⁿ⁻²⁾ ÷ a⁽⁴ⁿ⁻²⁾

= 1

4. Simplify using the law of indices.

(a) 2² × 4² ÷ 8²

= (2² × (2²)²) ÷ (2³)²

= (2² × 2⁴) ÷ 2⁶

= 2⁽²⁺⁴⁾ ÷ 2⁶

= 2⁶ ÷ 2⁶

= 1

(b) 5³ × 125³ ÷ 25³

= 5³ × (5³)³ ÷ (5²)³

= 5³ × 5⁹ ÷ 5⁶

= 5⁽³⁺⁹⁻⁶⁾

= 5⁶

(c) 4⁴ × 5⁵ ÷ 25³ × 16²

= (2²)⁴ × 5⁵ ÷ (5²)³ × (2⁴)²

= 2⁸ × 5⁵ ÷ 5⁶ × 2⁸

= 2⁸ ÷ 2⁸ × 5⁽⁵⁻⁶⁾

= 1 ÷ 5

= 1/5

5. Fill the box with appropriate number.

6. Prove that

(a) (𝑥^m+n+2 x 𝑥^m+n+2) / 𝑥^2(m+n+1)= 𝑥²

L.H.S:

(𝑥^m+n+2 x 𝑥^m+n+2) / 𝑥^2(m+n+1)

= (𝑥^m+n+2 x 𝑥^m+n+2) / 𝑥^2(m+n+1)

= 𝑥^{(m+n+2+m+n+2)} / 𝑥^2m+2n+2

= 𝑥^2m+n+4 / 𝑥^2m+2n+2

= 𝑥^{2m+2n+4−2m−2n−2}

= 𝑥² = R.H.S

(b) 𝑥 ^p-q+1 x  𝑥 ^q-r+1 x  𝑥 ^r-p+1 / 𝑥³= 1

L.H.S:

= 𝑥^p-q+1 x 𝑥^q-r+1 x 𝑥^r-p+1) / 𝑥³

= 𝑥^{p-q+1+q-r+1+r-p+1}) / 𝑥³

= 𝑥³) / 𝑥³

= 𝑥^3-3

= 𝑥⁰

= 1

= R.H.S

(c) 𝑥 ^{(a-b) (a-b)} x  𝑥 ^{(b-c) b-c} x  𝑥 ^{(c-a) c+a}= 1

L.H.S:

= 𝑥^{(a-b) a+b} x 𝑥^{(b-c) b-c} x 𝑥^{(c-a) c+a}

= 𝑥^{(a - b + a - b + b - c + b - c + c - a + c + a)}

= 𝑥⁰

= 1= R.H.S

If a = 2, b = 3 , c = 1, m = 4 and n = 5, find the value of

(a) a^m x bⁿ x c^ab / m^a x n^b x (b^{a )c}

= 2⁴ x 3⁵ x 1^{2 x 3} / 4² x 5³ x (3^{2 )1}

= 2⁴ x 3⁵ x 1⁶ / 2⁴ x 5³ x (3²

= 2⁴ x 3² x 3³ x 1⁶ / 2⁴ x 5³ x 3²

= (3/5)³

(b) (a + b + c)^{m + n} ÷ (m + n)^{a + b+ c}

= (2 + 3 + 1)^{4 + 5} / (4 + 5)^{2 + 3+ 1}

= 6⁹ / 9⁶