Rewrite the radical as a rational exponent. the cube root of 2 to the seventh power.

Solution:

Given, the cube root of 2 to the seventh power.

We have to write the radical as a rational exponent.

The formula to change a radical to a non-integer rational exponent. is given by

\(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\)

Now, cube root of 2 to the seventh power can be written as \(\sqrt[3]{2^{7}}\)

So, \(\sqrt[3]{2^{7}}=2^{\frac{7}{3}}\)

Therefore, the rational exponent is \(2^{\frac{7}{3}}\).

Example:

Rewrite the radical as a rational exponent. the cube root of 3 to the fourth power.

Solution:

Given, the cube root of 3 to the fourth power.

We have to write the radical as a rational exponent.

The formula to change a radical to rational exponent is given by

\(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\)

Now, cube root of 3 to the fourth power can be written as \(\sqrt[3]{3^{4}}\)

So, \(\sqrt[3]{3^{4}}=3^{\frac{4}{3}}\)

= 3

Therefore, the rational exponent is \(3^{\frac{4}{3}}\).

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Rewrite the radical as a rational exponent. the cube root of 2 to the seventh power.

Summary:

The radical cube root of 2 to the seventh power as a rational exponent \(2^{\frac{7}{3}}\).