Evaluating powers
Generally, it’s hard to work with both really large and really small numbers. Writing out huge numbers takes time, and that’s where powers step in! They help us rewrite a number in shorter form without losing its numerical value.
What does it mean to evaluate powers?
A power is an expression that represents repeated multiplication:
In the expression:
The number $$a$$ is called the base and $$b$$ is called the power. The power is also sometimes called the exponent.
Generally, to evaluate means to determine the equivalent numerical value of an initial expression.
Why is evaluating powers so useful?
Evaluating powers has many applications. One of the more commonly known examples is in scientific notation, which is used in many branches of science.
For example, you’re given the number $$0.000000000021$$. Just looking at it, you can tell that this number is really small; to copy it, you first have to count the zeros. You could make an error in counting and copying the wrong number. This number would be easier to copy if it could be written shorter. Luckily for us, it can be! This number can be written using powers as $$21\times 10^{-12}$$.
How to evaluate powers
Now that we know what powers are and why they’re useful, it’s time to see them in action! But, before that, let’s take a look at some properties of powers that can help us.
| Product of powers property | $$a^m\cdot a^n=a^{m+n}$$ |
| Quotient of powers property | $$\dfrac{a^m}{a^n}=a^{m-n}$$ |
| Power of a power property | $$\left(a^{x} \right)^{y}=a^{x y}$$ |
| Power of a product property | $$a^n\times b^n=(ab)^n$$ |
| Power of a quotient property | $$\dfrac{a^n}{b^n}=\left(\dfrac{a}{b}\right)^n$$ |
Now let’s walk through a problem together.
Example 1
Evaluate the power:
The property of raising the product to power states that you need to raise each factor to that same power $$(a\times b)^{n}=a^{n}\times b^{n}$$. Use this property and square the first number $$6$$ and the second number $$10^{-3}$$ separately:
$$6^2 \times (10^{-3})^{2}$$
Since for any non-zero real number $$a$$ and integers $$x$$ and $$y$$ is true $$\left(a^{x} \right)^{y}=a^{x\cdot y}$$, multiply the exponents $$-3$$ and $$2$$:
$$6^2 \times 10^{-3\cdot 2}$$
Multiply the numbers:
$$6^2 \times 10^{-6}$$
Square number $$6$$:
$$36 \times 10^{-6}$$
Write the expression as a product with the factor $$3.6$$ to rewrite the number in scientific notation.
$$3.6 \times 10^1 \times 10^{-6}$$
Multiply the terms with the same base by adding their exponents. Then add the numbers $$1$$ and $$-6$$.
Example 2
Evaluate the powers:
For any non-zero real number $$a$$ and any rational numbers $$n$$ and $$m$$ it follows: $$\frac{a^n}{a^m}=a^{n-m}$$. Since we have the quotient of powers, rewrite the base $$5$$ and subtract the exponent of the denominator from the exponent of the numerator:
$$5^{10-3}$$
Subtract the numbers:
That wasn’t so bad, right? Now that we’ve walked through detailed examples, let’s review the overall process so you can learn how to use it with any problem:
Study summary
- Use properties of powers to simplify the expression.
- State the solution or, if needed, write the number in scientific notation.
Try it yourself!
Practicing math concepts like this one is a great way to prepare yourself for the math journey to come! So, when you’re ready, we’ve got some practice problems for you!
📝 Evaluate the powers:
- $$6^3 \times 6^7$$
- $$(6 \times 10^{-3})^{2}\times(2 \times 10^{5})^4$$
- $$\frac{6^5}{2^4}\times3^7$$
- $$3^{-4} \times 9^2$$
Solutions:
- $$6^{10}$$
- $$5.76 \cdot10^{16}$$
- $$2\cdot3^{12}$$
- $$1$$
If you’re still struggling with the solving process, that’s totally okay! Stumbling a few times is good for the learning process. If you get too stuck or lost, scan the problem using your Photomath app and we’ll walk you through it!
Here’s a sneak peek of what you’ll see:
