Multiplying Monomials

After studying this lesson, you will be able to:

• Multiply monomials.
• Multiply powers with the same base.
• Raise a power to a power.

Monomials have one term. The term can be a number, a variable, or the product of a number and a variable. Monomials are expressions with do not contain a + or - sign.... it has only one term.

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Examples of monomials: 4, 4x, 4x ², -3x ²y, xyz

We will learn several exponent rules. It is very important that you learn these immediately.

In the example 5 ², 5 is the Base and 2 is the Power or Exponent

Multiplying Powers with the Same Base:

The base stays the same; Add exponents

Example 1

8 ⁴ · 8 ³

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We are multiplying powers with the same base (8). So, we keep the base and we add the exponents.

The answer will be 8 ⁷

Example 2

y · y ² · y ⁵

We are multiplying powers with the same base (y). So, we keep the base and we add the exponents.

The answer will be y ⁸

Example 3

x ² · x ⁴ · x

We are multiplying powers with the same base (x). So, we keep the base and we add the exponents.

The answer will be x ⁷

Example 4

( x ² y ³ )( x y ² )

We are multiplying powers with the same base. Actually, we have two bases (x and y). So, we keep the bases and we add the exponents. We add the exponents of x (2 and 1) and we add the exponents of y (3 and 2) to get the x ³ y ⁵

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Example 5

( a ³ b )( a ² b)( a ⁴ )

We are multiplying powers with the same base. Actually, we have two bases (a and b). So, we keep the bases and we add the exponents. The answer will be a ⁹ b ²

Now we're going to work with expressions that have coefficients. Remember that coefficients are the numbers in front of the variables. When multiplying powers with the same base, we add the exponents. We also multiply the coefficients.

Example 6

( 5x )( 2x ² )

The coefficients are 5 and 2 so we multiply those. The exponents of x are 1 and 2 so we add those to get the answer 10x ³

Example 7

( -3y ² )( -4y )

The coefficients are -3 and -4 so we multiply those. The exponents of y are 2 and 1 so we add those to get the answer 12 y ³

Example 8

( -2x ² y ³ z ⁴)( -xz )

The coefficients are -2 and -1 so we multiply those. The exponents of x are 2 and 1, the exponent of y is 3, and the exponents of z are 4 and 1 so we add those to get the answer 2 x ³ y ³ z ⁵

Example 9

The coefficients are -5, 3, and so we multiply those. The exponents of x are 2, 2, and 1 and the exponents of y are 2 and 4 so we add those to get the answer -6x ⁵ y ⁶