Question

simplify each of the following .
$({9}^{2}) ^{4}$
${(h}^{2}) {}^{5}$
$\frac{ {14}^{13} }{ {7}^{13} }$
$\frac{ {(9}^{5} ){}^{4} }{ {3}^{20} }$
​

Answer 1

$\sf\huge\bold{Answer:}$

$\fbox{Solution 1}$

$\sf = ({9}^{2}) ^{4}$

$\sf\purple{:⟶(a {}^{m} ) {}^{n} = a {}^{m \times n} }$

$\sf = {9}^{2 \times 4}$

$\sf = {9}^{8}$

$\sf = 43,046,721$

$\sf\huge\purple{ 43,046,721 }$

$\fbox{Solution 2}$

$\sf = {(h}^{2}) {}^{5}$

$\sf\purple{:⟶(a {}^{m} ) {}^{n} = a {}^{m \times n} }$

$\sf = {h}^{2 \times 5}$

$\sf = h {}^{10}$

$\sf\huge\purple{h {}^{10} }$

$\fbox{Solution 3}$

$\sf = \frac{ {14}^{13} }{ {7}^{13} }$

$\sf\purple{:⟶(ab) {}^{m} = a {}^{m} \times b {}^{m} }$

$\sf = \frac{(7 \times 2) {}^{13} }{ {7}^{13} }$

$\sf = \frac{ {7}^{13} \times {2}^{13} }{ {7}^{13} }$

$\sf{=}$$\displaystyle{\sf { \cancel{ \frac{7¹³}{7¹³} }}}$$\sf ×2¹³$

$\sf = {2}^{13}$

$\sf = 8,192$

$\sf\huge\purple{8,192 }$

$\fbox{Solution 4 }$

$= \frac{ {(9}^{5} ){}^{4} }{ {3}^{20} }$

$\sf\purple{:⟶(a {}^{m} ) {}^{n} = a {}^{m \times n} }$

$= \frac{9 {}^{5 \times 4} }{ {3}^{20} }$

$= \frac{9 {}^{20} }{ {3}^{20} }$

$\sf\purple{:⟶(ab) {}^{m} = a {}^{m} \times b {}^{m} }$

$= \frac{(3 \times 3) {}^{20} }{ {3}^{20} }$

$= \frac{3 {}^{20} \times {3}^{20} }{ {3}^{20} }$

$\sf{=}$$\displaystyle{\sf { \cancel{ \frac{3 {}^{20}}{3 {}^{20}} }}}$$\sf × 3 {}^{20}$

$\sf = {3}^{20}$

$\sf = 3,486,784,401$

$\sf\huge\purple{ 3,486,784,401}$