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Fraction Exponents,
Fraction Exponent to Radical


The exponents in Algebra page showed examples of whole number exponents exponents, and the effect they have on numbers and terms.

But it’s not just whole numbers that can be exponents, we can also have fraction exponents, often referred to as rational exponents.

We can also convert from fraction to radical form with terms involving fraction exponents.




Whole Number Exponent Recap:

With positive whole number exponents, they tell us how many times to multiply a number by itself.

32  =  3 × 3  =  9       ,       32  =  3 × 3 × 3  =  27       and so on.



Fraction Exponents:

Things work slightly different when fraction exponents are involved.

The easiest cases to look at first are when the numerator of a fraction exponent is the number 1.
Such fraction exponents in sums tell us what root of a number to find.

{\large{a}}^{\frac{1}{n}}   =   {\large{\sqrt[n]{a}}}

So for the number 3, we have:  

{\large{3}}^{\frac{1}{2}}   =   {\large{\sqrt[2]{3}}}   =   {\large{\sqrt{3}}}

{\large{3}}^{\frac{1}{3}}   =   {\large{\sqrt[3]{3}}}

{\large{3}}^{\frac{1}{3}}   =   {\large{\sqrt[4]{3}}}


So  {\large{a}}^{\frac{1}{n}}  means taking the  nth  root of  a.   Going from fraction to radical form.



Examples    


(1.1) 

{\large{36}}^{\frac{1}{2}} \space = \space {\large{\sqrt{3}}} \space = \space 6    ( 3 × 3 = 9 )


(1.2) 

{\large{64}}^{\frac{1}{3}} \space = \space {\large{\sqrt[3]{64}}} \space = \space 4    ( 4 × 4 × 4 = 64 )


(1.3) 

{\large{16}}^{\frac{1}{4}} \space = \space {\large{\sqrt[4]{16}}} \space = \space 2    ( 2 × 2 × 2 × 2 = 16 )


(1.4) 

{\large{{\text{-}}27}}^{\frac{1}{3}} \space = \space {\large{\sqrt[3]{{\text{-}}27}} \space = \space {\text{-}}3}    ( -3 × -3 × -3 = -27 )


(1.5) 

{\large{{\text{-}}81}}^{\frac{1}{4}} \space = \space {\large{\sqrt[4]{{\text{-}}81}}}

Here, there is no real number that can be raised to an exponent of 4 and give –16 as a result.
So we can’t evaluate the radical form here, this does happen sometimes.






Fraction Exponents,
Fraction Exponent to Radical Form Further


For further fraction exponents, when the numerator on top can be other numbers than 1.

We have:

{\large{a}}^{\frac{m}{n}} \space = \space {\large{\sqrt[n]{a^m}}}    or    ({\large{\sqrt[n]{a}}})^m.

As,

{\large{a}}^{\frac{m}{n}} \space = \space ( {\large{a}}^m )^{\frac{1}{n}} \space = \space {\large{\sqrt[n]{a^m}}},     but also     {\large{a}}^{\frac{m}{n}} \space = \space ( {\large{a}}^{\frac{1}{n}} )^m \space = \space ({\large{\sqrt[n]{a}}})^m.


Example    


(2.1) 

a)   {\large{27}}^{\frac{2}{3}} \space = \space ({\large{\sqrt[3]{27}}})^2 \space = \space ({\large{3}})^2 \space = \space 9

b)   {\large{27}}^{\frac{2}{3}} \space = \space {\large{\sqrt[3]{27^2}}} \space = \space {\large{\sqrt[3]{729}}} \space = \space 9





Simplifying Fraction Form:

There are times when one will want to simplify the fraction/exponent form of a term where variables are involved.
Below are 3 typical examples of such situations.



Examples    


(3.1) 

Simplify   ({\large{9a}}^3)^{\frac{1}{2}}.

Solution   

({\large{9a}}^3)^{\frac{1}{2}} \space = \space {\large{9}}^{\frac{1}{2}}{\large{a}}^{3({\frac{1}{2}})} \space = \space {\large{\sqrt{9}a}}^{\frac{3}{2}} \space = \space {\large{3a}}^{\frac{3}{2}}



(3.2) 

Simplify   {\large{x}}^{\frac{1}{2}}{\large{x}}^{\frac{7}{2}}.

Solution   

{\large{x}}^{\frac{1}{2}}{\large{x}}^{\frac{7}{2}} \space = \space {\large{x}}^{{\frac{1}{2}}+{\frac{7}{2}}} \space = \space {\large{x}}^{\frac{8}{2}} \space = \space {\large{x}}^4



(3.3) 

Simplify   ({\frac{r^2}{25t}}){\frac{1}{2}}.

Solution   

({\frac{r^2}{25t}}){\frac{1}{2}}   =   ({\frac{r^{2({\frac{1}{2}})}}{25^{\frac{1}{2}}t^{\frac{1}{2}}}})   =   ({\frac{\sqrt{r^2}}{\sqrt{25}\sqrt{t}}})   =   {\frac{r^2}{5\sqrt{t}}}


Now when simplifying an expression, we prefer to not have any radical signs in a fraction denominator.

Here multiplying the fraction we have by  {\frac{\sqrt{t}}{\sqrt{t}}}  will do this.

{\frac{r^2}{5\sqrt{t}}} × {\frac{\sqrt{t}}{\sqrt{t}}}   =   {\frac{r^2{\sqrt{t}}}{5t}}





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