# Rewrite without rational exponents, and simfly, if possible 256^{frac{3}{4}}

Question
Rewrite without rational exponents, and simfly, if possible $$\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}$$

2021-02-10
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 $$\displaystyle{256}^{{{\frac{{{3}}}{{{4}}}}}}={64}$$

### Relevant Questions

Rewrite without rational exponents, and simfly, if possible $$\displaystyle{\left({a}^{{{4}}}\ {b}^{{{4}}}\right)}^{{{\frac{{{1}}}{{{7}}}}}}$$
Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.
a. $$2\ln4x^{3}\ +\ 3\ln\ y\ -\ \frac{1}{3}\ln\ z^{6}$$
b. $$\ln(x^{2}\ -\ 16)\ -\ \ln(x\ +\ 4)$$
Rewrite each expression so that each term is in the form $$kx^{n},$$ where k is a real number, x is a positive real number, and n is a rational number.
a)$$x^{-\frac{2}{3}}\ \times\ x^{\frac{1}{3}}=?$$
b) $$\frac{10x^{\frac{1}{3}}}{2x^{2}}=?$$
c) $$(3x^{\frac{1}{4}})^{-2}=?$$
Solve the examples below. The answer should contain only positive indicators without fractional indicators in the denominator. Leave Rational (Not Radical) Answers:
$$1) (343b^{3})^\frac{4}{3}$$
$$2) (216x^{6})^{-\frac{5}{3}}$$
$$3) 3a^{-2}b^{\frac{5}{4}}\times 2a^{-\frac{7}{4}}b^{\frac{4}{3}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$
$$\displaystyle{5}{x}^{{{\frac{{{5}}}{{{2}}}}}}+{2}{x}^{{{\frac{{{1}}}{{{2}}}}}}+{x}^{{{\frac{{{3}}}{{{2}}}}}}$$
$$a) x^\frac{5}{2} - 9x^{\frac{1}{2}}$$
$$b) x^{-\frac{2}{3}}+2x(-\frac{1}{2})+x^{\frac{1}{2}}$$
$$\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}$$
$$\displaystyle{\frac{{{x}^{{{4}}}{\left({3}{x}\right)}^{{{2}}}}}{{{x}^{{{3}}}}}}$$