How can I rewrite the cube root of 2 raised to the seventh power as a rational exponent?

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Rewriting the Expression

To rewrite the given expression the cube root of 2 to the seventh power as a rational exponent, we’ll follow these steps:

Step 1: Understand the Fractional Exponent

The cube root of a number can be expressed as a rational exponent. In mathematical terms, the cube root of any number x is given by:

x^{1/3}

Thus, we can express the cube root of 2 as:

2^{1/3}

Step 2: Raise to the Seventh Power

Since we’re dealing with the cube root of 2 raised to the seventh power, we take our expression from Step 1 and raise it to the seventh power:

(2^{1/3})^{7}

Step 3: Apply the Power of a Power Rule

Using the rule of exponents that states (a^{m})^{n} = a^{m*n}, we can simplify our expression:

(2^{1/3})^{7} = 2^{(1/3) * 7} = 2^{7/3}

Final Result

Therefore, the cube root of 2 raised to the seventh power can be rewritten as:

2^{7/3}

This representation not only makes the expression easier to work with, but it also keeps it in a format that is compatible with various mathematical operations.

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