Question

simplify. \\(\sqrt{25y^{14}}\\) assume that the variable \\(y\\) represents a positive real number.

Explanation:

Step1: Simplify the square root of 25

The square root of 25 is 5, since \( \sqrt{25} = 5 \).

Step2: Simplify the square root of \( y^{14} \)

Using the property of exponents \( \sqrt{a^n} = a^{\frac{n}{2}} \) (for \( a \geq 0 \)), for \( y^{14} \), we have \( \sqrt{y^{14}} = y^{\frac{14}{2}} = y^7 \) (since \( y \) is a positive real number, we don't need to consider absolute value).

Step3: Combine the results

Using the property \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) (for \( a \geq 0, b \geq 0 \)), we get \( \sqrt{25y^{14}} = \sqrt{25} \cdot \sqrt{y^{14}} = 5 \cdot y^7 = 5y^7 \).

Answer:

\( 5y^7 \)