Question

write the following expression in simplest form.
$sqrt{2^{5} cdot 13^{3}}$
$sqrt{2^{5} cdot 13^{3}} = square$
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Simplify exponents inside square root

We know that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) and \(\sqrt{x^n}=x^{\frac{n}{2}}\). For \(2^5\), we can write \(2^5 = 2^{4 + 1}=2^4\times2^1\) and for \(13^3=13^{2+1}=13^2\times13^1\). So \(\sqrt{2^5\times13^3}=\sqrt{2^4\times2\times13^2\times13}\)

Step2: Apply square root property

Using \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we get \(\sqrt{2^4\times2\times13^2\times13}=\sqrt{2^4}\times\sqrt{2}\times\sqrt{13^2}\times\sqrt{13}\)
We know that \(\sqrt{x^m}=x^{\frac{m}{2}}\), so \(\sqrt{2^4} = 2^{\frac{4}{2}}=2^2 = 4\), \(\sqrt{13^2}=13^{\frac{2}{2}} = 13\)
Then the expression becomes \(4\times13\times\sqrt{2\times13}\)

Step3: Multiply the constants

Multiply \(4\) and \(13\): \(4\times13 = 52\) and \(\sqrt{2\times13}=\sqrt{26}\)
So the simplified form is \(52\sqrt{26}\)

Answer:

\(52\sqrt{26}\)