Sure, let me walk you through the steps to simplify the expression [tex]\( y = \frac{4}{\sqrt[4]{x^3}} - 5 \sqrt{x} \)[/tex].
1. Understanding the power notation:
- [tex]\(\sqrt[4]{x^3}\)[/tex] is the fourth root of [tex]\(x^3\)[/tex]. In terms of exponents, this can be written as [tex]\((x^3)^{1/4}\)[/tex].
- [tex]\(\sqrt{x}\)[/tex] is the square root of [tex]\(x\)[/tex], which can be written as [tex]\(x^{1/2}\)[/tex].
2. Simplifying the first term:
- Rewrite [tex]\(\sqrt[4]{x^3}\)[/tex] as [tex]\((x^3)^{1/4}\)[/tex].
- This simplifies further:
[tex]\[
(x^3)^{1/4} = x^{3/4}.
\][/tex]
- Thus, the first term becomes:
[tex]\[
\frac{4}{\sqrt[4]{x^3}} = \frac{4}{x^{3/4}}.
\][/tex]
3. Combining the exponents for the denominator:
- Using the properties of exponents, we know that [tex]\(\frac{1}{x^{3/4}} = x^{-3/4}\)[/tex].
- So, the first term further simplifies to:
[tex]\[
\frac{4}{x^{3/4}} = 4x^{-3/4}.
\][/tex]
4. Second term:
- The second term is already in a simplified exponent form, [tex]\( -5 \sqrt{x} = -5 x^{1/2} \)[/tex].
5. Combining both terms:
- Now, our expression is fully written in exponent form as follows:
[tex]\[
y = 4x^{-3/4} - 5x^{1/2}.
\][/tex]
Putting everything together, the simplified expression for [tex]\( y = \frac{4}{\sqrt[4]{x^3}} - 5 \sqrt{x} \)[/tex] becomes:
[tex]\[
y = -5\sqrt{x} + \frac{4}{(x^3)^{1/4}}.
\][/tex]
This step-by-step simplification gives us a clearer understanding of how the terms are combined to form the final expression:
[tex]\[
-5\sqrt{x} + 4/(x^3)^{1/4}.
\][/tex]
