Step 1

Rational exponents (also called fractional exponents) are expressions with exponents that are rational numbers (as opposed to integers).

Given rational exponent is \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}\), we have find answer without rational exponent.

Step 2

We know that

\(\displaystyle{256}={2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\)

\(\displaystyle{256}={2}^{{{8}}}\)

We can write \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}\) as

\(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}={\left({2}^{{{8}}}\right)}^{{{\frac{{{3}}}{{{4}}}}}}={2}^{{{\frac{{{8}\ \times\ {3}}}{{{4}}}}}}\)

\(\displaystyle={2}^{{{2}\ \times\ {3}}}={2}^{{{6}}}\)

\(\displaystyle={2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\)

\(\displaystyle={4}\ \times\ {4}\ \times\ {4}\)

\(\displaystyle{\left({256}\right)}^{{{\frac{{{3}}}{{{4}}}}}}={64}\)

So \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}={64}\)

Rational exponents (also called fractional exponents) are expressions with exponents that are rational numbers (as opposed to integers).

Given rational exponent is \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}\), we have find answer without rational exponent.

Step 2

We know that

\(\displaystyle{256}={2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\)

\(\displaystyle{256}={2}^{{{8}}}\)

We can write \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}\) as

\(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}={\left({2}^{{{8}}}\right)}^{{{\frac{{{3}}}{{{4}}}}}}={2}^{{{\frac{{{8}\ \times\ {3}}}{{{4}}}}}}\)

\(\displaystyle={2}^{{{2}\ \times\ {3}}}={2}^{{{6}}}\)

\(\displaystyle={2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\ \times\ {2}\)

\(\displaystyle={4}\ \times\ {4}\ \times\ {4}\)

\(\displaystyle{\left({256}\right)}^{{{\frac{{{3}}}{{{4}}}}}}={64}\)

So \(\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}={64}\)