Simplify and express the result in power notation with positive exponent.
(i) $(- 4)^{5} \div (- 4)^{8}$

(ii) $(\frac{1}{2 ^{3}})^{2}$

(iii) $(- 3)^{4} \times (\frac{5}{3})^{4}$

(iv) $(3^{- 7} \div 3^{- 10}) \times 3^{- 5}$

(v) $2^{- 3} \times (- 7)^{(- 3)}$

Question

Verified by Toppr

Answer 1

(i) $(- 4)^{5} \div (- 4)^{8} = \frac{( - 4 ) ^{5}}{( - 4 ) ^{8}} = (- 4)^{5 - 8} = (- 4)^{- 3} = \frac{1}{( - 4 ) ^{3}}$

(ii) $(\frac{1}{2 ^{3}})^{2} = \frac{1}{2 ^{3 \times 2}} = \frac{1}{2 ^{6}}$

(iii) $(- 3)^{4} \times (\frac{5}{3})^{4} = (- 1)^{4} \times (3)^{4} \times (\frac{5}{3})^{4} = (- 1)^{4} \times 5^{4} = 5^{4}$

(iv) $(3^{- 7} \div 3^{- 10}) \times 3^{- 5} = (\frac{3 ^{- 7}}{3 ^{- 10}}) \times (3)^{- 5} = 3^{- 7 + 10} \times 3^{- 5}$
$= 3^{3 - 5}$
$= 3^{- 2} = \frac{1}{3 ^{2}}$

(v) $2^{- 3} \times (- 7)^{(- 3)} = \frac{1}{2 ^{3}} \times \frac{1}{( - 7 ) ^{3}} = (\frac{1}{2 \times ( - 7 )})^{3}$
$= \frac{1}{( - 1 4 ) ^{3}}$

Answer 2

(i)

(- 4)^{5} \div (- 4)^{8} = \frac{( - 4 ) ^{5}}{( - 4 ) ^{8}} = (- 4)^{5 - 8} = (- 4)^{- 3} = \frac{1}{( - 4 ) ^{3}}

(ii)

(\frac{1}{2 ^{3}})^{2} = \frac{1}{2 ^{3 \times 2}} = \frac{1}{2 ^{6}}

(iii)

(- 3)^{4} \times (\frac{5}{3})^{4} = (- 1)^{4} \times (3)^{4} \times (\frac{5}{3})^{4} = (- 1)^{4} \times 5^{4} = 5^{4}

(iv)

(3^{- 7} \div 3^{- 10}) \times 3^{- 5} = (\frac{3 ^{- 7}}{3 ^{- 10}}) \times (3)^{- 5} = 3^{- 7 + 10} \times 3^{- 5}

= 3^{3 - 5}

= 3^{- 2} = \frac{1}{3 ^{2}}

(v)

2^{- 3} \times (- 7)^{(- 3)} = \frac{1}{2 ^{3}} \times \frac{1}{( - 7 ) ^{3}} = (\frac{1}{2 \times ( - 7 )})^{3}

= \frac{1}{( - 1 4 ) ^{3}}