Questions: Simplify the following radical expression. √[7]64 x^3

Solution Steps

To simplify the given radical expression \(\sqrt[7]{64 x^{3}}\), we need to express the radicand (the expression inside the radical) in terms of powers that can be simplified under the given root. We will look for perfect seventh powers within the expression and simplify accordingly.

We start with the expression \( \sqrt[7]{64 x^{3}} \). We can rewrite \( 64 \) as \( 2^6 \), so the expression becomes: \[ \sqrt[7]{2^6 x^{3}} \]

Using the property of radicals, we can separate the expression: \[ \sqrt[7]{2^6} \cdot \sqrt[7]{x^{3}} \] Now, we simplify each part. The seventh root of \( 2^6 \) can be expressed as: \[ 2^{\frac{6}{7}} \] And for \( x^{3} \): \[ \sqrt[7]{x^{3}} = x^{\frac{3}{7}} \] Thus, we have: \[ \sqrt[7]{64 x^{3}} = 2^{\frac{6}{7}} \cdot x^{\frac{3}{7}} \]

Combining the results, we get: \[ \sqrt[7]{64 x^{3}} = 2^{\frac{6}{7}} x^{\frac{3}{7}} \approx 1.8114 \cdot x^{\frac{3}{7}} \]

Final Answer