To simplify the expression [tex]\(\sqrt{y^{12}}\)[/tex], we need to consider the properties of exponents and roots. Here’s a detailed, step-by-step solution:
1. Understand the expression under the square root:
[tex]\[
\sqrt{y^{12}}
\][/tex]
2. Recall the property of exponents and roots: The square root of an expression can be rewritten using fractional exponents. Specifically:
[tex]\[
\sqrt{x} = x^{1/2}
\][/tex]
3. Apply the property to the given expression:
[tex]\[
\sqrt{y^{12}} = (y^{12})^{1/2}
\][/tex]
4. Use the exponent multiplication rule: When raising an expression with an exponent to another exponent, multiply the exponents:
[tex]\[
(y^{12})^{1/2} = y^{12 \cdot 1/2}
\][/tex]
5. Simplify the exponents:
[tex]\[
y^{12 \cdot 1/2} = y^6
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{y^{12}}\)[/tex] is:
[tex]\[
y^6
\][/tex]
This is the final simplified result:
[tex]\[
\sqrt{y^{12}} = y^6
\][/tex]
