Simplify and express the result in power notation with positive e… - Vidyadip

Question

Simplify and express the result in power notation with positive exponent.
(i) $(-4)^5+(-4)^8$ $\qquad$ (ii) $\left(\frac{1}{2^3}\right)^2$ $\qquad$
(iii) $(-3)^4 \times\left(\frac{5}{3}\right)^4$ $\qquad$ (iv) $\left(3^{-7}+3^{-10}\right) \times 3^{-5}$ $\qquad$
(v) $2^{-3} \times(-7)^{-3}$

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Answer

(i) We have, $(-4)^5+(-4)^8$
$
\begin{array}{l}
=\frac{(-4)^5}{(-4)^8}=\frac{1}{(-4)^8 \times(-4)^{-5}} \quad\left[\because a^m=\frac{1}{a^{-m}}\right] \\
=\frac{1}{(-4)^{8-5}}=\frac{1}{(-4)^3} \quad\left[\because a^m \times a^n=a^{m+n}\right]
\end{array}
$
which is the required form.
(ii) We have, $\left(\frac{1}{2^3}\right)^2$
$
\begin{array}{l}
=\frac{(1)^2}{\left(2^3\right)^2} \qquad{\left[\because\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\right]} \\=\frac{1}{2^6}\qquad{\left[\because\left(a^m\right)^n=a^{m \times n}\right]}
\end{array}
$
which is the required form.
(iii) We have, $(-3)^4 \times\left(\frac{5}{3}\right)^4$
$=(-1 \times 3)^4 \times\left(\frac{5}{3}\right)^4 \qquad[\because-a=-1 \times a]$
$=(-1)^4 \times 3^4 \times \frac{5^4}{3^4}$
$\left[\because(a \times b)^m=a^m \times b^m,\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\right]$
$=1 \times 5^4=(5)^4 \qquad\left[\because(-1)^4=1\right]$
which is the required form.
(iv) Ans. $\frac{1}{3^2}$
(v) Ans. $\left(\frac{-1}{14}\right)^3$

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