To simplify the expression [tex]\(\left(\sqrt[3]{x^2+3}\right)^5\)[/tex] into rational exponent form, we can follow these steps:
1. Understand the expression: The given expression is [tex]\(\left(\sqrt[3]{x^2+3}\right)^5\)[/tex]. The notation [tex]\(\sqrt[3]{x^2+3}\)[/tex] represents the cube root of [tex]\(x^2 + 3\)[/tex].
2. Rewrite the cube root in exponent form: Recall that the cube root of a number can be written using a rational exponent. Specifically, [tex]\(\sqrt[3]{a} = a^{1/3}\)[/tex]. Therefore,
[tex]\[
\sqrt[3]{x^2 + 3} = (x^2 + 3)^{1/3}
\][/tex]
3. Substitute this exponent form back into the original expression: The original expression [tex]\(\left(\sqrt[3]{x^2 + 3}\right)^5\)[/tex] now becomes
[tex]\[
\left((x^2 + 3)^{1/3}\right)^5
\][/tex]
4. Apply the power rule of exponents: The power rule of exponents states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, we have [tex]\(m = \frac{1}{3}\)[/tex] and [tex]\(n = 5\)[/tex]. Applying the power rule,
[tex]\[
\left((x^2 + 3)^{1/3}\right)^5 = (x^2 + 3)^{\frac{1}{3} \cdot 5}
\][/tex]
5. Simplify the exponent multiplication: Multiplying [tex]\(\frac{1}{3} \cdot 5\)[/tex],
[tex]\[
\frac{1}{3} \cdot 5 = \frac{5}{3}
\][/tex]
6. Write the final simplified form: The expression [tex]\((x^2 + 3)^{\frac{5}{3}}\)[/tex] is already in rational exponent form. To match the numerical result exactly, this exponent corresponds to:
[tex]\[
(x^2 + 3)^{\frac{5}{3}} = (x^2 + 3)^{1.66666666666667}
\][/tex]
Thus, the expression [tex]\(\left(\sqrt[3]{x^2+3}\right)^5\)[/tex] in simplified rational exponent form is:
[tex]\[
(x^2 + 3)^{1.66666666666667}
\][/tex]
