Dividing Monomials

After studying this lesson, you will be able to:

• Divide monomials.
• Simplify expressions with negative exponents.

Dividing Powers with the Same Base: The base stays the same; Subtract the exponents

After subtracting the exponents, you will put the remaining exponent where the largest exponent was to begin with. For example, if you have which is a division problem, we will subtract the exponents 5 -2 which gives us 3. We will be left with x ³ . We leave this in the numerator, since the largest exponent was in the numerator to begin with.

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Example 1

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We are dividing powers with the same base (x). So, we keep the base and we subtract the exponents. We will have x to the 2 ^nd power and we put that in the denominator since the larger exponent was in the denominator to begin with.

The answer will be . We have to put a 1 in the numerator to hold the place. You can't leave the numerator without anything there.

Example 2

We are dividing powers with the same base (y). So, we keep the base and we subtract the exponents. We will have y to the 3 ^rd power and we put that in the numerator since the larger exponent was in the numerator to begin with.

The answer will be or y ³ . We don't have to put a 1 in the denominator.

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Example 3

We are dividing powers with the same base (a and b). So, we keep the bases and we subtract the exponents

The answer will be a ³ b.

Now we're going to work with expressions that have coefficients. Remember that coefficients are the numbers in front of the variables. When dividing powers with the same base, we subtract the exponents. We also divide or reduce the coefficients.

Example 4

The coefficients are -6 and 18 so we either divide or reduce those. In the case, we need to reduce the coefficients. The exponents of r are 2 and 2 so we subtract those. The exponents of s are 5 and 1 so we subtract those.

This will give us an answer of s because -6 and 18 reduces to -1 and 3. When we subtract the exponents of r we find that they cancel each other out. When we subtract the exponents of s, we are left with s ⁴ . We put that in the numerator since the larger exponent was in the numerator to begin with.

Example 5

The coefficients are 14 and 10 so we either divide or reduce those. In the case, we need to reduce the coefficients. 14 and 10 will reduce to 7 and 5. The exponents of x are 2 and 4 so we subtract those. The exponents of y are 1 and 1 so we subtract those. The x ² will go in the denominator and the y ² will go in the numerator.

This will give us an answer of