Certainly! Let's transform the expression [tex]\(\left(\sqrt[4]{x^7}\right)^3\)[/tex] into an expression with rational exponents step by step.
1. Express the fourth root as a fractional exponent:
The fourth root of [tex]\(x^7\)[/tex] can be written as [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{7}{4}\)[/tex].
[tex]\[
\sqrt[4]{x^7} = x^{\frac{7}{4}}
\][/tex]
2. Consider the whole expression [tex]\(\left(\sqrt[4]{x^7}\right)^3\)[/tex]:
Since we need to find [tex]\(\left(x^{\frac{7}{4}}\right)^3\)[/tex], we will raise [tex]\(x^{\frac{7}{4}}\)[/tex] to the power of 3.
3. Use the rules of exponents to combine the powers:
When you raise a power to a power, you multiply the exponents. So we have:
[tex]\[
\left(x^{\frac{7}{4}}\right)^3 = x^{\frac{7}{4} \cdot 3}
\][/tex]
4. Simplify the exponent multiplication:
Multiply [tex]\(\frac{7}{4}\)[/tex] by 3:
[tex]\[
\frac{7}{4} \cdot 3 = \frac{7 \cdot 3}{4} = \frac{21}{4}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\left(\sqrt[4]{x^7}\right)^3\)[/tex] is:
[tex]\[
x^{\frac{21}{4}}
\][/tex]
Or as a decimal, it is:
[tex]\[
x^{5.25}
\][/tex]
This final result shows the original expression simplified to a single term with a rational exponent.
