# Exponents with Multiplication and Division

When multiplying and dividing exponents, only the bases have to be the same in order to combine like terms. When like bases are multiplied, you will add the exponents. For example, 3^{2} x 3^{3} could be written 3^{5} (3x3x3x3x3). When dividing exponents with like bases, you will subtract the exponents. For example, 2^{6} ÷ 2^{4} = 2^{2} (2x2x2x2x2x2 ÷ 2x2x2x2). When raising a power to another power, for example (5^{2})^{3}, you will multiply the exponents. So, (5^{2})^{3} = 5^{2} x 5^{2} x 5^{2} = 5^{6}.

## Multiplication and Division with Exponents

When multiplying and dividing, only the bases have to be the same in order to combine like terms. When like bases are multiplied- three would be our base here.

We’re multiplying 3 squared times 3 cubed. The bases are alike. That means we add our integer exponents. The reason for this is 3 squared means 3 times 3 times 3 cubed (3 times 3 times 3).

When you multiply 3 squared times 3 cubed, you end up with 3 times itself 1, 2, 3, 4, 5 times. It would be 3 to the fifth, which you could then simplify further if necessary.

All we had to do was add our exponents, 2 plus 3 and 3 squared times 3 cubed. 3 to the fifth. When dividing with like bases, we subtract our exponent. Here’s the reason for that.

If you have 2 to the 6th divided by 2 to the fourth, then that’s 2 times itself 6 times. 1, 2, 3, 4, 5, 6 divided by 2 times itself four times, then 2 divided by 2 is 1.

2 divided by 2 is one. 2 divided by 2 is 1. 2 divided by 2 is 1. You’re left with just 2 times itself two times, or 2 squared. All we had to do was subtract our exponents.

2 to the sixth divided by 2 to the fourth is 2 squared, which, again, you could simplify further: 2 squared is 4. Our last one is when raising a power to a power.

**When you raise a power to a power, you multiply the exponent.** The reason for that is what we’re doing here is 5 squared times itself three times, which would be 5 squared times 5 squared times 5 squared.

5 squared is 5 times itself two times, times 5 squared (5 times itself two times) times 5 squared (5 times itself two times), and you end up with 5 times itself 1, 2, 3, 4, 5, 6 times.

**All we had to do was multiply our exponents together.** 5 squared cubed is 5 to the sixth. When multiplying with like bases, we add our exponents. When dividing with like bases, we subtract the exponent. When raising a power to a power, we multiply the exponent.