Question

what is the rational exponent and the valueif the radical expression is ^5√7776

Answer 1

what's the rational exponent

Answer 2

A rational exponent is an exponent that can be expressed as a fraction, where the numerator indicates the power and the denominator indicates the root. In the case of a radical expression, the \( n \)-th root of a number can be expressed with a rational exponent as follows:

\[
\sqrt[n]{x} = x^{\frac{1}{n}}
\]

For the expression \( \sqrt[5]{7776} \), it can be written with a rational exponent as:

\[
7776^{\frac{1}{5}}
\]

To find the value of \( \sqrt[5]{7776} \), we need to determine what number, when raised to the 5th power, equals 7776.

Calculating \( 7776^{\frac{1}{5}} \):

1. Factor \( 7776 \) into its prime components:
- \( 7776 = 2^5 \times 3^5 \)

2. Using this factorization, we find the fifth root:
- \( \sqrt[5]{7776} = \sqrt[5]{2^5 \times 3^5} = 2 \times 3 = 6 \)

Thus, the value of \( \sqrt[5]{7776} \) is:

\[
\sqrt[5]{7776} = 6
\]

So the answer is:

\[
\text{Value of } \sqrt[5]{7776} = 6
\]