Question

which expression is equivalent to \\(\sqrt{64a^6}\\)?
\\(\bigcirc\\ 8|a^3|\\)
\\(\bigcirc\\ 8a^6\\)
\\(\bigcirc\\ \sqrt{8|a^3|}\\)
\\(\bigcirc\\ \sqrt{8a^6}\\)

Explanation:

Step1: Simplify the square root of 64

The square root of 64 is 8, since \( \sqrt{64} = 8 \).

Step2: Simplify the square root of \( a^6 \)

Using the property of exponents \( \sqrt{x^n}=|x|^{\frac{n}{2}} \) (for even roots to ensure non - negativity), for \( \sqrt{a^6} \), we have \( \frac{6}{2}=3 \), so \( \sqrt{a^6}=|a^3| \).

Step3: Combine the two results

By the property of square roots \( \sqrt{xy}=\sqrt{x}\cdot\sqrt{y} \) (where \( x = 64 \) and \( y=a^6 \)), we get \( \sqrt{64a^6}=\sqrt{64}\cdot\sqrt{a^6}=8|a^3| \).

Answer:

A. \( 8|a^3| \)