Question

simplify \\(\sqrt{y^4}\\)

Explanation:

Step1: Recall square - root property

The square root of a number (or a variable expression) \( \sqrt{a^n} \) can be simplified using the property of exponents. For a non - negative real number \( a \) and positive integer \( n \), \( \sqrt{a^n}=a^{\frac{n}{2}} \) when \( n \) is even (since we are dealing with the principal square root, we assume \( a\geq0 \) or the expression is defined in the real - number system).

In the given expression \( \sqrt{y^{4}} \), here \( a = y \) and \( n = 4 \).

Step2: Apply the exponent rule

Using the formula \( \sqrt{a^n}=a^{\frac{n}{2}} \), substitute \( a = y \) and \( n = 4 \) into it. We get \( \sqrt{y^{4}}=y^{\frac{4}{2}} \).

Simplify the exponent \( \frac{4}{2}=2 \). So \( y^{\frac{4}{2}}=y^{2} \). Also, since \( (y^{2})^{2}=y^{4} \), the square root of \( y^{4} \) is \( y^{2} \) (for real numbers, if we consider the principal square root, and assuming \( y \) is a real number, \( \sqrt{y^{4}}=\vert y^{2}\vert=y^{2} \) because \( y^{2}\geq0 \) for all real \( y \)).

Answer:

\( y^{2} \)