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Simplify Expressions with Rational Exponents

A Rational Exponent is an exponent which is in the form of a fraction. In the expression a^n, a is the base and n is the exponent. If n is a fraction, it is called Rational Exponent.

You will be able to learn how to simplify expressions with rational exponents after going the following topics. 

1. How to convert radical form to exponent form
2. Properties of Rational Exponents
3. How to use the properties to simplify expressions
4. Examples of simplifying expressions


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Conversion of Radical Form to Exponent Form

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Every radical form can be converted into exponent form as shown in the below tabular form. 

Examples:

  Radical Form   Exponent Form
$\sqrt{x}$ $x^{\frac{1}{2}}$
$\sqrt[3]{x}$ $x^{\frac{1}{3}}$
$\sqrt[4]{x}$ $x^{\frac{1}{4}}$
$\sqrt[n]{x}$ $x^{\frac{1}{n}}$

In the above table, the Exponents of the terms in the Exponent Form column are in a fraction form and hence they are Rational Exponents.

Properties of Rational Exponents

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To simply the expressions with Rational exponents, the following rules are used. They are also called as Laws of Exponents. 

Exponent Properties

1. Product of Powers  $x^a\ .\ x^b$ = $x^{(a\ +\ b)}$

2. Power to a Power   $(x^a)^b$ = $x^{(a\ . \ b)}$

3. Quotient os Powers $x^a\ \div\ x^b$ = $x^{(a\ -\ b)}$

4. Power of Product   $(xy)^a$ = $x^a y^a$

5. power of Quotient  $(x\ \div y)^a$ = $x^a\ \div y^a$

6. Negative Exponent  $x^{-a}$ = $1\ \div x^a$

7. Zero Exponent      $x^0$ = $1$ 

Examples related to rules:
  • $2^3\ .\ 2^4$ = $2^{3+4}$ = $128$
  • $2^5\ /\ 2^3$ = $2^{5\ -\ 3}$ = $4$
  • $(2^3)^2$ = $2^{3.2}$ = $64$
  • $3^2\ .\ 4^2$ = $(3.4)^2$ = $144$
  • $4^3\ /\ 2^3$ = $(4/2)^3$ = $8$
  • $2^{-3}$ = $1/2^3$ = $0.125$
  • $5^0$ = 1
Apart from the above rules, there are two important properties of radicals which are useful in simplifying expressions. They are 

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

Simplifying Expressions with Rational Exponents using the Rules

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The below simple examples show how to simplify expressions with rational exponents.
a. Example:   Evaluate $27^{\frac{2}{3}}$.

Solution:
      $27^{\frac{2}{3}}$ = $(3^3)^{\frac{2}{3}}$ = $3^2$ = $9$
b. Example:  Write the below expression in simplest form.  $\frac{6xy^\frac{1}{2}}{2x^{\frac{1}{3}}z^{-5}}$

Solution: $\frac{6xy^\frac{1}{2}}{2x^{\frac{1}{3}}z^{-5}}$$3x^{(1\ -\ \frac{1}{3})} y^{\frac{1}{2}} z^{-(-5)}$ = $3x^{\frac{2}{3}} y^{\frac{1}{2}}z^{5}$
c. Example: Simplify the following expression.  $\sqrt[3]{\frac{x}{y^7}}$

Solution:

$\sqrt[3]{\frac{x}{y^7}}$ = $\sqrt[3]{\frac{xy^2}{y^7y^2}}$     Make the denominator a perfect cube.
              
 = $\sqrt[3]{\frac{xy^2}{y^9}}$          Simply.
     
 = $\frac{\sqrt[3]{xy^2}}{\sqrt[3]{y^9}}$    Quotient Property.

 = $\frac{\sqrt[3]{xy^2}}{y^3}$             Simplify.
d. Example:  Evaluate $-64^{\frac{1}{3}}$ 

Solution:

$-64^{\frac{1}{3}}$ = - $\sqrt[3]{64}$= $-4$

Examples of Simplifying Expressions

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Example 1:

Simplify: $\frac{10\ b^2\ c^2}{c \sqrt[3]{8b^4}}$

Solution:

$\frac{10\ b^2\ c^2}{c · \sqrt[3]{8} · \sqrt[3]{b^4}}$    Separate the factors in the denominator

$\frac{10\ b^2\ c^2}{c · 2 · \sqrt[3]{b^4}}$  Take the cube root of 8, which is $2$
 
$\frac{10\ b^2\ c^2}{c · 2 · b{^\frac{4}{3}}}$         Rewrite the radical using a fractional exponent

$\frac{10}{2}· \frac{c^2}{c} · \frac{b^2}{b^{\frac{4}{3}}}$      Rewrite the fraction as a series of factors in order to cancel factors 

$5 · c ·$ $\frac{b^2}{b^{\frac{4}{3}}}$      Simplify the constant and $c$ factors

$5 · c ·$ $b^{2-\frac{4}{3}}$      Use the rule of negative exponents 

$5 · c ·$ $b^{\frac{2}{3}}$                  Combine the b factors by adding the exponents 

$5\ c$ $\sqrt[3]{b^2}$                   

Change the expression with fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator
Example 2:
Simplify the below expression  $(\frac{25\ x^{-6}y(z^{-11})^2}{5(x^{-2})^5y^8z^2})^{-2}$

Solution:

Step 1: Take the reciprocal of the fraction and make the outside exponent positive.

$(\frac{5(x^{-2})^5 y^8 z^2}{25x^{-6} y(z^{-11})^2})^2$

Step 2: Get rid of any inside parentheses.

$(\frac{5x^{-10} y^8 z^2}{25x^{-6} yz^{-22}})^2$

Step 3: Reduce any fractional coefficients. Since no term has fractional coefficients, it remains same. 

$(\frac{5x^{-10} y^8 z^2}{25x^{-6} yz^{-22}})^2$

Step 4: Move all negatives either up or down. Make the exponents positive

$(\frac{5x^6 y^8 z^2 z^{22}}{25x^{10} y})^2$

Step 5: Combine all like bases by applying rules of exponents

$(\frac{5x^{6\ -\ 10} y^{8\ -\ 1} z^{2\ +\ 22}}{25})^2$

Step 6: Simplify the powers

$(\frac{x^{-4} y^7 z^{24}}{5})^2$

Step 7: Distribute the power to all exponents and simplify

$\frac{y^{14}\ z^{48}}{25\ x^8}$

This is the simplified expression
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