Question

which expression is equivalent to \\( \sqrt{64a^6} \\)?
\\( 8|a^3| \\)
\\( 8a^6 \\)
\\( \sqrt{8|a^3|} \\)
\\( \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 \( x = a \) and \( n = 6 \), we have \( \sqrt{a^6}=a^{\frac{6}{2}} = a^3 \)? Wait, no, actually, when dealing with square roots, we consider the principal square root, and for even exponents, \( \sqrt{a^{2k}} = |a^k| \), but if we assume \( a \) is non - negative (or in a context where absolute value is not necessary for simplification, like if \( a\geq0 \)), then \( \sqrt{a^6}=a^3 \)? Wait, no, \( (a^3)^2=a^6 \), but also \( (-a^3)^2=a^6 \). However, the square root of a non - negative number is non - negative. So \( \sqrt{64a^6}=\sqrt{64}\times\sqrt{a^6} \). We know that \( \sqrt{64} = 8 \), and \( \sqrt{a^6}=a^3 \) if we consider the principal square root (assuming \( a\) is such that \( a^3\geq0 \), or in a simplified form without absolute value for the sake of this problem). Wait, but let's check the options. The fourth option is \( \sqrt{8a^6} \)? No, wait the original problem is \( \sqrt{64a^6} \). Let's re - calculate:

\( \sqrt{64a^6}=\sqrt{64}\times\sqrt{a^6} \). Since \( \sqrt{64} = 8 \) and \( \sqrt{a^6}=a^3 \) (if we take the principal square root, and for the purpose of matching the options, let's see the options:

Option 1: \( 8|a^3| \)

Option 2: \( 8a^6 \)

Option 3: \( \sqrt{8|a^3|} \)

Option 4: \( \sqrt{8a^6} \)

Wait, no, let's do the calculation correctly. \( \sqrt{64a^6}=\sqrt{64}\times\sqrt{a^6} \). We know that \( \sqrt{64}=8 \), and \( \sqrt{a^6}=|a^3| \) because \( (a^3)^2=a^6 \), and the square root of a number squared is the absolute value of the number. So \( \sqrt{64a^6}=8|a^3| \)

Step3: Match with the options

Looking at the options, the first option is \( 8|a^3| \), which matches our result.

Answer:

\( 8|a^3| \) (the first option)