Sure, let's simplify the given expression [tex]\( \sqrt[6]{y^3} \)[/tex] by expressing it with radicals.
### Step-by-Step Solution:
1. Identify the Radical Expression:
The expression given is [tex]\( \sqrt[6]{y^3} \)[/tex].
2. Understand the Relationship Between Radicals and Exponents:
A general rule for radicals is: [tex]\(\sqrt[n]{a^b}\)[/tex] can be expressed as [tex]\(a^{\frac{b}{n}}\)[/tex].
3. Apply the Rule:
Here, we have [tex]\(a = y\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(n = 6\)[/tex]. Plugging these values into the rule, we get:
[tex]\[
\sqrt[6]{y^3} = y^{\frac{3}{6}}
\][/tex]
4. Simplify the Fraction:
Simplify the fraction [tex]\(\frac{3}{6}\)[/tex]. The greatest common divisor (GCD) of 3 and 6 is 3, so dividing both the numerator and the denominator by 3, we get:
[tex]\[
y^{\frac{3}{6}} = y^{\frac{1}{2}}
\][/tex]
5. Express Result using Radicals:
The exponent [tex]\(\frac{1}{2}\)[/tex] can be written back in radical form as the square root:
[tex]\[
y^{\frac{1}{2}} = \sqrt{y}
\][/tex]
### Final Simplified Expression:
[tex]\[
\sqrt[6]{y^3} = \sqrt{y}
\][/tex]
Therefore, the simplified form of [tex]\( \sqrt[6]{y^3} \)[/tex] is [tex]\( \sqrt{y} \)[/tex].
