Sure! Let's simplify the given expressions step-by-step.
a.) Simplification of [tex]\(\left(\frac{w^{-2}}{16 v^{1 / 2}}\right)^{1 / 4}\)[/tex]
1. Start with the given expression:
[tex]\[
\left(\frac{w^{-2}}{16 v^{1 / 2}}\right)^{1 / 4}
\][/tex]
2. Rewrite [tex]\( w^{-2} \)[/tex] as [tex]\( \frac{1}{w^{2}} \)[/tex]:
[tex]\[
\left(\frac{\frac{1}{w^{2}}}{16 v^{1 / 2}}\right)^{1 / 4}
\][/tex]
3. Simplify the fraction:
[tex]\[
\left(\frac{1}{16 v^{1 / 2} w^{2}}\right)^{1 / 4}
\][/tex]
4. Apply the fourth root to both the numerator and the denominator:
[tex]\[
\frac{1^{1/4}}{(16 v^{1/2} w^{2})^{1/4}}
\][/tex]
Since [tex]\( 1^{1/4} = 1 \)[/tex]:
[tex]\[
\frac{1}{(16 v^{1/2} w^{2})^{1/4}}
\][/tex]
5. Split the fourth root across the terms inside the denominator:
[tex]\[
\frac{1}{16^{1/4} \cdot (v^{1/2})^{1/4} \cdot (w^{2})^{1/4}}
\][/tex]
6. Simplify each component separately:
- [tex]\( 16^{1/4} = 2 \)[/tex] (because [tex]\( 16 = 2^4 \)[/tex])
- [tex]\( (v^{1/2})^{1/4} = v^{(1/2) \cdot (1/4)} = v^{1/8} \)[/tex]
- [tex]\( (w^{2})^{1/4} = w^{2 \cdot (1/4)} = w^{1/2} \)[/tex]
So, the simplified form is:
[tex]\[
\frac{1}{2 \cdot v^{1/8} \cdot w^{1/2}}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{1}{2 \sqrt[8]{v} \sqrt{w}}
\][/tex]
b.) Simplification of [tex]\(\frac{10}{\sqrt{x^4+y^5}}\)[/tex]
1. Start with the given expression:
[tex]\[
\frac{10}{\sqrt{x^4 + y^5}}
\][/tex]
2. Notice that the expression [tex]\( \sqrt{x^4 + y^5} \)[/tex] inside the denominator does not simplify directly into a more elementary form.
Thus, the simplified form remains:
[tex]\[
\frac{10}{\sqrt{x^4 + y^5}}
\][/tex]
These are the simplified forms of the given expressions:
a.) [tex]\(\frac{1}{2 \sqrt[8]{v} \sqrt{w}}\)[/tex]
b.) [tex]\(\frac{10}{\sqrt{x^4 + y^5}}\)[/tex]
