To solve the expression [tex]\(\sqrt{a b^{\frac{1}{4}}}\)[/tex], let's break it down into simpler parts and proceed step-by-step.
1. Start with the expression inside the square root: [tex]\(a b^{\frac{1}{4}}\)[/tex].
2. Understand that [tex]\(b^{\frac{1}{4}}\)[/tex] represents the fourth root of [tex]\(b\)[/tex]. So, [tex]\(b^{\frac{1}{4}}\)[/tex] is the number which, when raised to the power of 4, equals [tex]\(b\)[/tex].
3. Now, you then need to multiply [tex]\(a\)[/tex] by [tex]\(b^{\frac{1}{4}}\)[/tex]. This results in [tex]\(a b^{\frac{1}{4}}\)[/tex].
4. The next step is to take the square root of this entire product. The square root of [tex]\(a b^{\frac{1}{4}}\)[/tex] can be expressed as [tex]\(\left(a b^{\frac{1}{4}}\right)^{\frac{1}{2}}\)[/tex].
5. To simplify this expression, use the properties of exponents. Specifically, [tex]\(\left(x^m\right)^n = x^{mn}\)[/tex].
6. Apply this property:
[tex]\[\left(a b^{\frac{1}{4}}\right)^{\frac{1}{2}} = a^{\frac{1}{2}} \left(b^{\frac{1}{4}}\right)^{\frac{1}{2}}\][/tex]
7. Simplify [tex]\(\left(b^{\frac{1}{4}}\right)^{\frac{1}{2}}\)[/tex]:
[tex]\[\left(b^{\frac{1}{4}}\right)^{\frac{1}{2}} = b^{\left(\frac{1}{4} \cdot \frac{1}{2}\right)} = b^{\frac{1}{8}}\][/tex]
8. Now combine the results:
[tex]\[\sqrt{a b^{\frac{1}{4}}} = a^{\frac{1}{2}} b^{\frac{1}{8}}\][/tex]
Thus, [tex]\(\sqrt{a b^{\frac{1}{4}}}\)[/tex] simplifies to [tex]\(a^{\frac{1}{2}} b^{\frac{1}{8}}\)[/tex].
