Exponents: Power rule: (a^x)^y = a^(xy)

Properties of Exponents:

In mathematics, dealing with exponents is a regular occurrence. It is for this reason that we need quick, easy-to-use tricks in order to effectively and efficiently work with exponents. Fortunately, there are many simple tricks available, tricks that we call "properties of exponents".

Of all of the properties, otherwise known as laws or rules, "power rule" is arguably the most important law of exponents to know.

What is the Power Rule?

Power Rule, or Power Law, is a property of exponents that is defined by the following general formula:

$(a^x)^y=a^{x \cdot y}$

In words, the above expression basically states that for any value to an exponent, which is then all raised to another exponent, you can simply combine the exponents into one by just multiplying them. This is often just referred to as "raising a power to a power". For a video explanation, check out the tutorial on power of a power rule.

The best way to illustrate this concept is to put it into practice.

Let's do some examples of taking a power of a power with the power rule.

Evaluate $(2^3)^4$

For this first example, let's keep things very simple. In this case, we have an expression similar to that of the general formula above. The only work we have to do to solve this problem is multiplying powers. After that, we can solve!

$(2^3)^4=2^{3 \cdot 4} = 2^{12} = 4096$

Evaluate $((2^3)^4)^2$

This example is just an expansion on the previous one. In this case however, it is important to recognize that we do not have to use the power rule with the third exponent "2". We can solve this problem in one of two ways:

i) $((2^3)^4)^2=(2^{3 \cdot 4})^2=(2^{12})^2=(4096)^2=16777216$

ii) $((2^3)^4)^2=2^{3 \cdot 4 \cdot 2} = 2^{24} =16777216$

Either of these options works perfectly well, and it is really up to you to decide how you want to solve the problem.

Evaluate $(9a^4)^2$

In this example, we're making this a little more difficult by bringing in a variable with a constant, all of which is raised to another power. We are still taking the power of a power, but we need to do an extra step to deal with the variable and constant.

To solve, all we need to remember is to apply the exponent "2" to both the constant "9" as well as a4 using the power rule. If we remember to do all this, solving this problem is no issue at all!

$(9a^4)^2=(9^2) \cdot a^{4 \cdot 2}=81a^8$

Evaluate $(\frac{3a^4}{5b^{10}c^7})^2$

For this last example, let's tackle a much more complicated problem. This problem may seem daunting at first, but using what we've learned from power rule, it will end up being no problem at all to solve it. For a video solution to this problem, click the link here.

For this problem, solving it is very similar to that of example 3. We just need to apply the power "2" to every term in the brackets as follows:

After we do that, we just need to evaluate every term and solve!

And that's all there is to it! Now you should be well suited to solve any problem using power rule. For more practice this with law, and with other properties of exponents, look here.

Lastly, if you're looking for help on power rule derivative questions, we've got you covered in our Calculus course.