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Rational Exponents


Exponents play an important role in algebra and are used to express roots and it tells us how many times a number can be used in multiplication.
Example : 10$^{2}$, 10 squared
15$^{2}$, 15 squared
35$^{2}$, 35 squared and so on.
Exponents helps in writing repeated multiplication of the same number as seen above.
Expression that uses roots such as square root, cube root is known as a radical. It is important to note the length of radicals.

A fractional exponent is of the form x$^{\frac{a}{b}}$ where x is the radicand, a and b are exponents and index respectively.
a and b are positive integers and x$^{\frac{a}{b}}$ is a real number. Operations that are applied on exponents with non fractional powers can be performed on fractional exponents. Instead of integers we work on rational exponents.
When performing mathematical operations on radical exponents such as add or subtract during multiplication or division just find a common denominator to add or subtract numerators.

Rational exponent is of the form
x$^\frac{a}{b}$ = $\sqrt[b]{x^{a}}$, 'b' should not be equal to zero.
If a rational exponent is of the form $\frac{a}{b}$ then,

  1. Either do the 'a' th power, then take the 'b'th root.
  2. Or take the 'b' th root and do the 'a' th power.

Laws of Rational Exponents

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Laws of rational exponents are given below:
  1. x$^{a}$.x$^{b}$ = x$^{a+b}$
  2. $\frac{x^{a}}{x^{b}}$ = x$^{a-b}$
  3. (x$^{a})^{b}$ = x$^{ab}$
  4. ($\frac{m}{n})^{x}$= $\frac{m^{x}}{n^{x}}$
  5. (mn)$^{x}$ = m$^{x}$.n$^{x}$
  6. m$^{-a}$ = $\frac{1}{m^{a}}$
  7. m$^{0}$ = 1
  8. m$^{1}$ = m

Examples of Rational Exponents

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Example1: Simplify: x$^\frac{3}{4}$ . x$^\frac{4}{3}$
Solution : Using the property of multiplying powers simplify the given expression.

x$^\frac{3}{4}$ . x$^\frac{4}{3}$

Least common multiple of the given exponents is 12.

= x$^{(\frac{9}{12})+(\frac{16}{12})}$

= x$^{\frac{25}{12}}$

Example2: Simplify : $\frac{135y^{\frac{1}{4}}}{15^{\frac{1}{4}}}$
= ($\frac{135y}{15})^{\frac{1}{4}}$)

= (9y)$^{\frac{1}{4}}$

Example 3: Simplify 5(5)$^{\frac{1}{5}}+3(5)^{\frac{1}{5}}$
Solution :
= (5 + 3) (5)$^{\frac{1}{5}}$

= 8 (5)$^{\frac{1}{5}}$

Example 5: Simplify (16)$^{-\frac{4}{5}}$
(16)$^{-\frac{4}{5}}$ = ($\frac{1}{16})^{\frac{4}{5}}$

= $\frac{1}{4^{2 * (\frac{4}{5})}}$

= $\frac{1}{4^{\frac{8}{5}}}$

= $4^{-\frac{8}{5}}$