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Fractions with Power!

We saw integers raised to powers, but what about fractions? Well, fractions aren't much different.

Remember that raising a number to a power means you multiply that number by itself that many times. So you'd be right to assume that if you raise a fraction to a power you multiply the fraction by itself that many times.

Large chunks of boring text are bland, so let's see an example. Suppose we wanted to find $\left(\frac23\right)^3$. $$\left(\frac23\right)^3=\frac23\cdot\frac23\cdot\frac23$$

The third power indicates we want to multiply the base by itself $3$ times. This is an exercise in multiplying fractions.

As a result, it can be simplified to: $$\frac23\cdot\frac23\cdot\frac23=\frac{2\cdot2\cdot2}{3\cdot3\cdot3}=\frac{2^3}{3^3}$$ See what happened? The power of $3$ got distributed to both the numerator and denominator of the fraction. That wasn't too bad right? $$\frac{2^3}{3^3}=\frac{8}{27}$$

In general, the rule is:

This rule sometimes gets confused with fractional exponents. Those come up when we are dealing with square roots or cube roots. But here, the fraction is not the exponent. The fraction is the base.

Raising a Quotient to a Power

This image describes a common rule for exponents. It is used if we want to apply an exponent to an entire fraction. This means both its numerator and its denominator.

Let's do one more example.

We are trying to expand $\left(\frac{12}{4x^4}\right)^2.$ If we use the variables from the graphic, this means that $x=12, y=4x^4,$ and $a=2.$ Now all we have to do is substitute these values into our formula. Here are links for raising a product to a power and the power rule for exponents in case you need them. $$\begin{align} \left(\frac{x}{y}\right)^a&=\frac{x^a}{y^a}\\ \\ \left(\frac{12}{4x^4}\right)^2&=\frac{3}{x^4}\\ &=\frac{3^2}{(x^4)^2}\\ &=\frac{9}{(x^4)^2} \end{align}$$ Using the power rule for exponents we find that: $$\begin{align} &=\dfrac{9}{x^{4 \cdot 2}}\\ &=\boxed{\dfrac{9}{x^{8}}} \end{align}$$

(Video) Applying the Rules of Exponents

by PatrickJMT

Here are some great, more complex problems of raising a quotient to a power from Patrick JMT.

When raising a quotient or a fraction to a power, we raise both terms (the numerator and denominator) to the power. $$\left(\frac{m}{n}\right)^a=\frac{m^a}{n^a}\\ (m\div n)^a=m^a\div n^a$$

$\large{\text{Summary}}$

Let's work through some problems. The first is: $$\left(\frac{x^2}{2}\right)^4$$ We want to distribute the fourth power to both the numerator and denominator. $$\left(\frac{x^2}{2}\right)^4=\frac{(x^2)^4}{(2)^4}=\frac{x^{2\cdot4}}{2^4}=\boxed{\frac{x^8}{16}}$$

Another Example

The second problem is similar. $$\left(\frac{4}{n^6}\right)^2$$ Again, we want to distribute the square to both the numerator and denominator. $$\left(\frac{4}{n^6}\right)^2=\frac{(4)^2}{(n^6)^2}=\frac{4^2}{n^{6\cdot2}}=\boxed{\frac{16}{n^{12}}}$$

In the last step we used the power rule for exponents. It tells us that $(n^6)^2 = n^{6 \cdot 2} = n^{12}$. This became the denominator of the fraction above.