Question

1. Simplify the following using laws of exponents. (i) \( 2^{10} \times 2^{4} \) (ii) \( \left(3^{2}\right) \times\left(3^{2}\right)^{4} \) (iii) \( \frac{5^{7}}{5^{2}} \) (iv) \( 9^{2} \times 9^{18} \times 9^{10} \) (v) \( \left(\frac{3}{5}\right)^{4} \times\left(\frac{3}{5}\right)^{3} \times\left(\frac{3}{5}\right)^{8} \) (vi) \( (-3)^{3} \times(-3)^{10} \times(-3)^{7} \) (vii) \( \left(3^{2}\right)^{2} \) (viii) \( 2^{4} \times 3^{4} \) (ix) \( 2^{4 a} \times 2^{5 a} \) (x) \( \left(10^{2}\right)^{3} \) (xi) \( \left[\left(\frac{-5}{6}\right)^{2}\right]^{5} \) (xii) \( 2^{3 a+7} \times 2^{7 a+3} \) (xiii) \( \left(\frac{2}{3}\right)^{5} \) (xiv) \( (-3)^{3} \times(-5)^{3} \) \( (x v) \frac{(-4)^{6}}{(-4)^{3}} \) (xvi) \( \frac{9^{7}}{9^{15}} \) (xvii) \( \frac{(-6)^{5}}{(-6)^{9}} \) (xviii) \( (-7)^{7} \times(-7)^{8} \) (xix) \( \left(-6^{4}\right)^{4} \) (xx) \( a^{x} \times a^{y} \times a^{z} \)

Answer 1

(i) 2¹⁰ × 2⁴ = 2¹⁰⁺⁴ = 2¹⁴

[∵ a^m × aⁿ = a^m+n]

(ii) (3²) × (3²)⁴ 

 (iii) \(\frac {5^7}{5^2}\) = 5^{7 – 2} = 5⁵

= 5 × 5 × 5 × 5 × 5 = 5⁵

[∵ \(\frac {a^m}{a^n}\)= a^m-n, m > n]

(iv) 9² × 9¹⁸ × 9¹⁰ = 9^2+18+10 = 9³⁰

[∵ a^m × aⁿ = a^m+n]

(v) (\((\frac 34)^4\times (\frac 35)^3 \times (\frac 35)^8\) =  \((\frac 35)^{4+3+8} = (\frac 35)^{15}\)

[∵ a^m × aⁿ = a^m+n]

(vi) (-3)³ × (-3)¹⁰ × (-3)⁷ = (-3)^{3 + 10 + 7} = (-3)²⁰

[∵ a^m × aⁿ = a^m+n]

(vii) 3(²)² = 3^2×2 = 3⁴

[∵ (a^m)ⁿ = a^mn])

(viii) 2⁴ × 3⁴ = (2 × 3 )⁴ = 6⁴

[∵ a^m × b^m = (ab)^m]

(ix) 2^4a × 2^5a = 2^4a+5a = 2^9a

[∵ a^m × aⁿ = a^m+n]

(x) (10²)³ = 10^2×3 = 10⁶
[∵ (a^m)ⁿ = a^m×n ]

(xiv) (-3)³ x (-5)³

= (-3 x -5) = 15³