Certainly! Let's convert the expression [tex]\(\sqrt[4]{7 x^{-3}}\)[/tex] into exponential form step by step.
Step 1: Understand the expression inside the radical
- You have the expression [tex]\(7 x^{-3}\)[/tex].
Step 2: Recall the properties of radicals
- The fourth root of a number [tex]\(a\)[/tex] is written as [tex]\(a^{1/4}\)[/tex]. Using this property, we can rewrite [tex]\(\sqrt[4]{a}\)[/tex] as [tex]\(a^{1/4}\)[/tex].
Step 3: Apply the property of radicals to the entire expression inside the radical
- We rewrite the expression [tex]\(\sqrt[4]{7 x^{-3}}\)[/tex] as [tex]\((7 x^{-3})^{1/4}\)[/tex].
Step 4: Distribute the exponent of [tex]\(1/4\)[/tex] to both the 7 and the [tex]\(x^{-3}\)[/tex]
- Using the property of exponents [tex]\((a \cdot b)^c = a^c \cdot b^c\)[/tex], we can distribute the exponent: [tex]\((7)^{1/4} \cdot (x^{-3})^{1/4}\)[/tex].
Step 5: Simplify each part
- [tex]\(7^{1/4}\)[/tex] remains as is.
- [tex]\((x^{-3})^{1/4}\)[/tex] can be simplified using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\((-3) \cdot (1/4) = -3/4\)[/tex], so [tex]\((x^{-3})^{1/4} = x^{-3/4}\)[/tex].
Step 6: Combine the simplified parts
- Combining them gives us: [tex]\(7^{1/4} \cdot x^{-3/4}\)[/tex].
Thus, the exponential form of [tex]\(\sqrt[4]{7 x^{-3}}\)[/tex] is:
[tex]\[
7^{1/4} \cdot x^{-3/4}
\][/tex]
