How to simplify powers of fractions and fractional exponents
Powers on fractions vs. powers that are fractions
This lesson will cover how to find the power of a fraction as well as introduce how to work with fractional exponents.
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Part 1: Powers of fractions
Say we have something like
???\left(\frac{a}{b}\right)^c???
where ???a,b??? and ???c??? are integers. This is like saying that we’re multiplying ???a/b??? by itself ???c??? times. This turns the power problem into a fraction multiplication problem, where you multiply the numerators together and the denominators together. In the case of this example, ???a??? is the numerator and ???b??? is the denominator.
Part 2: Fractional powers with like bases
If we start with something like ???x^a \cdot x^{c/d}??? (where ???a, c??? and ???d??? are integers and ???x??? is a real number) we have like bases because the base of both terms is ???x???. When that’s the case, we add the exponents.
???x^a \cdot x^{\frac{c}{d}} = x^{a+\frac{c}{d}}???
Now the problem is just about fraction addition.
How to simplify a fraction raised to a power, or bases that are raised to fractions
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Examples with fractions and powers
Example
Simplify the expression.
???\left(\frac{3}{4}\right)^2???
This is an example of a power of a fraction. The way the problem is written, it’s like saying that we’re multiplying ???3/4??? by itself twice, since the base is ???3/4??? and the exponent is ???2???. So the problem becomes
???\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)???
Now we’ve got a fraction multiplication problem. When we multiply fractions, we multiply the numerators together, and we multiply the denominators together.
???\frac{3\cdot3}{4\cdot4}=\frac{9}{16}???
Let’s look at an example with variables.
When we multiply fractions, we multiply the numerators together, and we multiply the denominators together.
Example
Simplify the expression.
???\left(\frac{a}{b^3}\right)^4???
This is an example of a power of a fraction. The way the problem is written, it’s like saying that we’re multiplying ???a/b^3??? by itself four times, since the base is ???a/b^3??? and the exponent is ???4???. So the problem becomes
???\left(\frac{a}{b^3}\right)\left(\frac{a}{b^3}\right)\left(\frac{a}{b^3}\right)\left(\frac{a}{b^3}\right)???
Now we’ve got a fraction multiplication problem. Remember, when we multiply fractions, we multiply the numerators together and then we multiply the denominators together.
???\frac{a\cdot a\cdot a\cdot a}{b^3 \cdot b^3 \cdot b^3 \cdot b^3}???
Now we have a like base of ???a??? in the numerator and a like base of ???b??? in the denominator.
In the numerator we can write ???a^4??? because we have ???a??? multiplied by itself four times.
Remember when you have a like base you can add the exponents, we will need to do this for the denominator. Let’s look at the calculation for the denominator:
???b^3 \cdot b^3 \cdot b^3 \cdot b^3??? ???=??? ???b^{3+3+3+3}??? ???=??? ???b^{12}???
So the simplified expression is
???\frac{a^4}{b^{12}}???
Let’s look at an example with like bases.
Example
Simplify the expression.
???a^3 \cdot a^{\frac14}???
We have like bases because the base of both terms is ???a???. When that’s the case, we add the exponents.
???a^{3+\frac{1}{4}}???
Now the problem is just about fraction addition in the exponent. To add the fractions, we have to find a common denominator.
???a^{3(\frac{4}{4}) + \frac{1}{4}}???
???a^{\frac{12}{4}+\frac{1}{4}}???
???a^{\frac{13}{4}}???
