is a small number that appears in the upper right-hand corner of a base. Sometimes, an exponent is referred to as a power or index. This number says how many times to use the base in a multiplication. For example:
= (2 x 2 x 2) = 8
Exponents can be tricky to deal with, but only if you’re unfamiliar with them. Knowing the properties of exponents, and how to use them to rewrite expressions – like radical expressions
– can help you better navigate the TASC Test Assessing Secondary Completion™ Math subtest.
The Rules of Exponents
There are three keys to understanding exponents:
- The exponent says how many times to use the number in a multiplication. This, you’ll know, is the basic definition of exponents.
- A negative exponent means divide, because division is the opposite function of multiplying.
- A fractional exponent, like 1/x, means to take the xth root. For example: x1/2 = 2√x
If you understand these three keys, you can essentially work through any exponent because all the other rules of exponents are based on these. Keep this in mind as you look at the following properties of exponents.
You may be asked to solve an equation such as: 52
. You could solve for each exponent, and then multiply the two numbers. Or, you can take a shortcut. Because of the properties of exponents, you can simply add the exponents together. For the above example, the answer is: 510
When an equation asks you to multiply two powers with the same exponent but different bases, you should multiply the bases (like normal) and keep the exponent. For example:
Usually, you can leave answers in this exponent form. However, if you don’t notice this answer in a multiple-choice question then you should solve for the exponent. Remember, all you need to solve for the exponent is to multiply the number by itself as many times as indicated by the exponent.
Any base that is raised to the power of zero is 1. That means all of these exponents equal 1:
This may seem strange to you – especially because you know any number multiplied by zero equals zero. The complex answer is that for all real numbers x, x ≠ 0.
It should be noted that 00 is undefined
You may be asked to take the power of a power. In this case, you will multiply the exponent in the base by the exponent that is acting on the base. For example:
The steps are slightly different if you raise a product to a power. In this case, you are actually raising each factor to a power:
Just as you can multiple exponents, you can also divide exponents. When you divide powers with the same base, you simply subtract the exponents like this:
26 = (2 x 2 x 2 x 2 x 2 x 2) = 22
24 (2 x 2 x 2 x 2)
Here, we’ve crossed out a 2 for each 2 on the bottom to demonstrate what the subtraction of exponents actually looks like. Even more simply, you can subtract 4 from 6.
If you see a quotient raised to a power, you must distribute the exponent to both the numerator and the denominator:
Simplify Radical Expressions
Using what we know about the properties of exponents – such as the third key above and the quotient of powers property – we get the product property of radicals, and the quotient property of radicals. These two properties demonstrate that the square root of a product equals the product of the square roots of the factors
. For example:
√2x = (√2 x √x)
In radical expressions, the answer cannot be negative; we wouldn’t be able to solve for a real answer if it was.
When you simplify a radical expression, you know it is in its simplest form when there are:
- No perfect square factors other than 1 in the radicand (or division symbol)
- No fractions in the radicand
- No radicals in the denominator of a fraction
If the denominator is not a perfect square, then you can rationalize the denominator by multiplying the expression by an appropriate form of one. Check out examples of these radical expressions, and practice simplifying a radical expression