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.

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

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

---

Take the course

Want to learn more about Algebra 2? I have a step-by-step course for that. :)

---

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.

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

$a^{\frac{12}{4}+\frac{1}{4}}$

$a^{\frac{13}{4}}$

---

Get access to the complete Algebra 2 course

---