Question

use the law of rational exponent to simplify

1. (n4)3/2

2. 3b1/2 • b3/2

3. (a1/2)1/3

4. (12)3/4 / (4)1/4

5. (a3b4) 1/12


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Answer 1

The simplified form the following are :

1. $n^{6}$

2. $3b^{2}$

3. $a^{\frac{1}{6} }$

4. 3(12)$^{\frac{1}{2} }$

5. (a)$^{\frac{1}{4} }$(b)$^{\frac{1}{3} }$

Given :

The form needs to be simplified.

To Find :

The simplified form of the given form

Solution :

First, we need to know "The Laws of Rational Exponent"

Rational Exponents:  A rational exponent is expressed in the form of                               $x^{\frac{a}{b }$ a base as 'x' and the exponent as '$\frac{a}{b}$'.

The laws of Rational Exponents are:

• $a^{m}$ × $a^{n}$ = $a^{m+n}$  
•  $\frac{a^{m} }{a^{n} }$ = $a^{m-n}$

•  $(a^{m} )^{n}$ = $a^{mn}$

• $a^{m}$ + $b^{m}$ = $(ab)^{m}$

• $a^{-m}$ = $\frac{1}{a^{m} }$

1. $(n^{4)} ^{\frac{3}{2} }$  = $n^{\frac{12}{2} }$          (Using $(a^{m})^{n}$ = $a^{mn}$)

            = $n^{6}$

2. 3$(b^{\frac{1}{2} })(b^{\frac{3}{2} } )$ = 3$b^{2}$    ( Using $a^{m}$ +$a^{n}$ = $a^{m + n}$ )

3.  $(a^{\frac{1}{2} })^{\frac{1}{3} }$ = $a^{\frac{1}{6} }$           ( Using $(a^{m} )^{n}$ = $a^{mn}$ )

4. $\frac{12^{\frac{3}{4} } }{4^{\frac{1}{4} } }$  = $\frac{(4^{\frac{3}{4} } )(3^{\frac{3}{4} } ) }{4^{\frac{1}{4} } }$      ( Using $\frac{a^{m} }{a^{n} } = a^{m-n}$  )

          = $4^{\frac{1}{2} }3^{\frac{3}{4}$

          = 3$(12^{\frac{1}{2} } )$

5. $(a^{3} b^{4} )^{\frac{1}{12} }$ = $(a^{3} )^{\frac{1}{12} }$  ×  $(b^{4} )^{\frac{1}{12} }$    ( Using $(a^{m} )^{n} = (a^{mn} )$ )

                 =  $(a^{\frac{1}{4} } )(b^{\frac{1}{3} } )$

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