Question
Simplify \( \left(\sqrt[5]{7 x^{6}}\right)^{10} \) \( [?] x \)
Ask by Phillips Clark. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 49x^{12} \)
Solution
To simplify the expression:
\[
\left(\sqrt[5]{7 x^{6}}\right)^{10}
\]
**Step 1:** Rewrite the fifth root as an exponent.
\[
\left(7 x^{6}\right)^{\frac{1}{5}}
\]
**Step 2:** Apply the exponent outside the parentheses.
\[
\left(7 x^{6}\right)^{\frac{1}{5} \times 10} = \left(7 x^{6}\right)^{2}
\]
**Step 3:** Distribute the exponent to both the coefficient and the variable.
\[
7^{2} \times \left(x^{6}\right)^{2} = 49 x^{12}
\]
**Final Simplified Form:**
\[
49 x^{12}
\]
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Simplify this solution Extra Insights
To simplify \( \left(\sqrt[5]{7 x^{6}}\right)^{10} \), we can start by rewriting the expression using exponent rules. The fifth root can be expressed as an exponent of \( \frac{1}{5} \): \[ \left(7 x^{6}\right)^{\frac{10}{5}} = \left(7 x^{6}\right)^{2} \] Next, we can distribute the exponent to both the constant and the variable: \[ = 7^{2} \cdot (x^{6})^{2} \] Calculating \( 7^{2} \): \[ = 49 \] And for \( (x^{6})^{2} \), we multiply the exponents: \[ = x^{12} \] Combining these two results, we get: \[ = 49 x^{12} \] So, the simplified expression is \( 49 x^{12} \).
