Express the following with a single exponent:
${(2^{10})}^{3} \times {(2^{5})}^{4}$

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Solution

For any non-zero integer $a$ , where $m and n$ are whole numbers,
${(a^{m})}^{n} = a^{m \times n}$
So, ${(2^{10})}^{3}$ $=$ $2^{10 \times 3}$ where $a = 2, m = 10, n = 3$
$= 2^{30}$ …………………. $(1)$
Also, ${(2^{5})}^{4}$ $= 2^{5 \times 4}$ where $a = 2, m = 5, n = 4$
$= 2^{20}$ …………………. $(2)$
Therefore, ${(2^{10})}^{3} \times {(2^{5})}^{4}$ $= 2^{30} \times 2^{20}$ (From $(1)$ and $(2)$ )
we know that, $a^{m} \times a^{n} = a^{m + n}$
Exponents can be added only when the bases are the same.
So, $2^{30} \times 2^{20} = 2^{30 + 20}$ where $a = 2, m = 30, n = 20$
$= 2^{50}$ .
Therefore, ${(2^{10})}^{3} \times {(2^{5})}^{4}$ $= 2^{50}$ .

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