Monomials and Powers

Use the properties of exponents to multiply monomials. To multiply factors that have the same base, add their exponents. Multiply their coefficients.

Multiplying Monomials

Example

Simplify the expression (2x³)(-3x).

Practice 

Simplify the expression.

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2    

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You can use the distributive property and the properties of exponents to find the product of a monomial and a binomial.

Using the Distributive Property

Example

Simplify the expression 2n(4n² − 5).

Practice

Simplify the expression.

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5    

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You can use the rule below to simplify a power of a product.

To simplify a power of a product, find the power of each factor and multiply.

   (ab)^m = a^m • b^m       (5 • 2)³ = 5³ • 2³
      

Simplifying a Power of a Product

The radius of a container is twice its height. Write an expression for the volume of the container. Use the formula V = πr²h.

The radius is twice the height, so r = 2h.

An expression for the volume of the container is V = 4πh³.

Power of a Power Property
 
To simplify a power of a power, multiply exponents.

    (a^m)ⁿ = a^mn      (5³)² = 5^{3 • 2} = 5⁶
          

Simplifying a Power of a Power

Practice

Simplify the expression.

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8    

9  Match the expression with the rule used to simplify it.

10  Simplify the expression.

Step 1  Write the expression in expanded form.
Step 2  Simplify by multiplying numerators and multiplying denominators.
Step 3  Write a rule you could use to find the power of a quotient.

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12  You need fabric for a window seat cushion. Use the trapezoid pattern shown to write a polynomial expression for the area of the top of the cushion. Simplify the expression.

Answers

1.  -14m³

2.  -8x³

3.  4a³

4.  2t³ − 8t²

5.  5n⁴ − 3n²

6.  6x² − 8x

7.  a⁴b²

8.  25m⁶

9.  1-B, 2-A, 3-C

10.  $\frac{x^4}{y^4}$

11.  D

12.  2b² − 12b