By Brian D.

As with any exam that tests mathematical knowledge, you’re going to need to know the basics. These include topics like exponents, factors, etc. Here we will be doing a brief review of the basic information that you should know for this test, and many others in general.

### Laws of Exponents

Here is how exponents are typically represented in a problem.

a^{b}

a represents the **base** and b represents the **exponent**.

· The **base** is the number that is being multiplied.

· The **exponent** is the number of times the base is multiplied by itself.

If you have a negative exponent, then it simply means that you need to convert it to fractional form, in which the numerator is always 1 and the denominator is the base raised to the exponent with a positive sign rather than a negative sign.

If you have a base with a fractional exponent, then the first thing you do is put the base raised to the fraction’s numerator under a radical sign. The radical will be raised to the denominator’s root. So to sum it up, the numerator goes under the radical sign and the denominator goes outside.

If you’re presented with a situation in which you’re asked to multiple numbers that have the same base but different exponents, then what you do is add the exponents while keeping the base the same. On the other hand, if the exponents are the same but the bases are different, then you will not be able to multiple them this way. You’ll either have to use a calculator or use the specific circumstances to find a way short of manually multiplying the numbers (not only will that be time consuming but also very prone to mistakes).

The same above applies to division of bases with exponents. Instead of adding the exponents (if applicable) you subtract them.

If you have an entire fraction raised to an exponent, then you simply raise both the numerator and denominator to that exponent.

If you have a term nested in parentheses raised to a power, then you raise every term within those parenthesis raised to that power. For instance:

(XY)^{n} = X^{n}Y^{n}

But if there are no parenthesis, then you do not raise the entire term to that power

XY^{n}! = X^{n}Y^{n}

### Factor Trees

Factor trees are a visual method of determining factors and ultimately, prime factors. This is not recommended because it is time consuming. In the time used to draw an entire tree (especially for larger numbers), you could have been solving other problems that present a greater challenge or require more attention. You should use this only if you do not feel comfortable with (prime) factorization because you have the *multiple choice* working to your advantage, if you *plug in* rather than spending the time to do the problem the long way.

If you do decide to use factor trees, then you simply find factors of the specified number and keep on doing that until you have only prime numbers and 1. When nothing more can be further simplified, then you have your prime factors. If the number at the top of the tree cannot be further simplified however, then it is already a prime number.