Powers And Roots Of Fractions

I. Introduction

Suppose we have a fraction of the form $\left(\dfrac{a}{b}\right)^n$, then we can write, that

$\left(\dfrac{a}{b}\right)^n = \dfrac{(a)^n}{(b)^n}$

The same is true for roots. So, we can say that,

$\sqrt[\large{m}]{\dfrac{a}{b}} = \dfrac{\sqrt[\large{m}]{a}}{\sqrt[m]{b}}$

In terms of properties of operations, as described we can say $\unicode{0x201C}exponentiation\ is\ distributive\ over\ division\ and\ multiplication\unicode{0x201D}$

We will see the detailed reason for this in the following sections.

II. Distributive Over Multiplication

Let's consider a number which is a product of two numbers raised to the power some exponent, that is, a number of the form $(a \cdot b)^n$.

$(a \cdot b)^n$

$= a \cdot b \times a \cdot b \times a \cdot b \times ... (n\ times)$

Since there are $n$ $a$'s and $n$ $b$'s in the above expression, we can rearrange the $a$'s and $b$'s together to write the expression as:

$[a \times a \times a \times ... (n\ times)] \times [b \times b \times b \times ... (n\ times)]$

$= a^n \times b^n$

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III. Distributive Over Division

Now we will take a look at the distributive property of exponentiation over division.

Let us consider a number of the form $\left( \dfrac{a}{b} \right)^n$. Expanding this we can wite:

$\left( \dfrac{a}{b} \right)^n$

$= $

$\dfrac{a}{b} \times \dfrac{a}{b} \times \dfrac{a}{b} ... (n\ times)$

$= \dfrac{a \times a \times a ... (n\ times)}{b \times b \times b ... (n\ times)}$

$= \dfrac{a^n}{b^n}$

IV. Examples

We will apply these concepts to calculate a few examples.

$\underline{\text{Example 1:}}$

$\left(\dfrac{2}{3}\right)^3 =\ ?$

$\left(\dfrac{2}{3}\right)^3$

$=\dfrac{2^3}{3^3}$

$=\dfrac{8}{27}$

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$\underline{\text{Example 2:}}$

$\sqrt[4]{\dfrac{16}{81}} =\ ?$

$\sqrt[4]{\dfrac{16}{81}}$

$=\dfrac{\sqrt[4]{16}}{\sqrt[4]{81}}$

$=\dfrac{2}{3}$

Now, let us try the following problem:

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Introduction To Powers And Roots