Question

select the expression that is equivalent to \\(\sqrt{100x^6y^{10}}\\) given \\(x > 0\\) and \\(y > 0\\). answer \\(50x^3y^5\\) \\(10x^3y^5\\) \\(10x^{\frac{1}{2}}y^{\frac{1}{2}}\\) \\(50x^{\frac{1}{2}}y^{\frac{1}{2}}\\)

Explanation:

Step1: Simplify the square root of 100

The square root of 100 is 10, since \( \sqrt{100} = 10 \).

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

Using the property \( \sqrt{a^n} = a^{\frac{n}{2}} \) (for \( a>0 \)), for \( x^6 \), we have \( \sqrt{x^6} = x^{\frac{6}{2}} = x^3 \) (since \( x > 0 \)).

Step3: Simplify the square root of \( y^{10} \)

Similarly, for \( y^{10} \), \( \sqrt{y^{10}} = y^{\frac{10}{2}} = y^5 \) (since \( y > 0 \)).

Step4: Combine the simplified parts

Multiply the simplified parts together: \( 10 \times x^3 \times y^5 = 10x^3y^5 \).

Answer:

\( 10x^3y^5 \) (corresponding to the option "10\( x^3 \)\( y^5 \)")