Question

question
simplify \\(\sqrt{x^{20}}\\) completely given \\(x > 0\\).
answer attempt 1 out of 2

Explanation:

Step1: Recall the property of square roots and exponents

For a non - negative real number \(a\) and positive integer \(n\), \(\sqrt{a^{m}}=a^{\frac{m}{2}}\) when \(m\) is an even integer (since we are dealing with square root, the index is 2). Also, when \(x>0\), we can use the power - of - a - power rule \((a^{m})^{n}=a^{mn}\).

Given \(\sqrt{x^{20}}\), we can rewrite the square root as an exponent. The square root is the same as raising to the \(\frac{1}{2}\) power. So \(\sqrt{x^{20}}=(x^{20})^{\frac{1}{2}}\).

Step2: Apply the power - of - a - power rule

According to the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\), when \(a = x\), \(m = 20\) and \(n=\frac{1}{2}\), we have \((x^{20})^{\frac{1}{2}}=x^{20\times\frac{1}{2}}\).

Calculate \(20\times\frac{1}{2}=10\). So \((x^{20})^{\frac{1}{2}} = x^{10}\).

Answer:

\(x^{10}\)