Question

How do you simplify fractions with exponents?

Answer 1

Hint: In this we have to simplify the fractions with exponents, for this we have to know the rules and formulas related to exponents, and applying the exponents’ rules and formulas we can simplify the expression in fractions.

Complete step by step solution:
Exponents are defined as when an expression or a statement of specific natural numbers are represented as a repeated power by multiplication of its units then the resulting number is called as an exponent. The resulting set of numbers are the same as the original sequence.
Fraction of a whole: When we divide a whole into equal parts, each part is a fraction of the whole.
A fraction has two parts. The number on the top of the line is called the numerator. It tells how many equal parts of the whole or collection are taken. The number below the line is called the denominator. It shows the total divisible number of equal parts the whole into or the total number of equal parts which are there in a collection.
Fraction can be represented by \[\dfrac{{{\text{numerator}}}}{{{\text{denominator}}}}\],
Now we can only simplify fractions with exponents if either their bases or exponents are equal.
If bases are equal then you can write the fraction as one power using the formula:
\[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
If exponents are equal then you can use the formula:
\[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\] , and finally simplify the fraction \[\dfrac{a}{b}\] if possible.
Let’s take an example:
Simplify the expression \[{\left( {\dfrac{{16}}{{25}}} \right)^{\dfrac{3}{2}}}\],
Now transform the expression inside the power , we get,
\[ \Rightarrow {\left( {\dfrac{{{4^2}}}{{{5^2}}}} \right)^{\dfrac{3}{2}}}\],
Now taking out the common power we get,
\[ \Rightarrow {\left( {{{\left( {\dfrac{4}{5}} \right)}^2}} \right)^{\dfrac{3}{2}}}\],
Now using exponential identities, we get,
\[ \Rightarrow {\left( {\left( {\dfrac{4}{5}} \right)} \right)^{2 \times \dfrac{3}{2}}}\],
Now simplifying we get,
\[ \Rightarrow {\left( {\dfrac{4}{5}} \right)^3}\],
Now simplifying we get,
\[ \Rightarrow \dfrac{{{4^3}}}{{{5^3}}}\],
Now further simplification we get,
\[ \Rightarrow \dfrac{{64}}{{125}}\],
Now we can say,
\[ \Rightarrow {\left( {\dfrac{{16}}{{25}}} \right)^{\dfrac{3}{2}}} = \dfrac{{64}}{{125}}\].
So, to simplify the expression we will make use of exponential formulas and identities.

Note:
Negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. Only move the negative exponents.