Algebra II

The following are rules regarding the multiplying of variable expressions.

• Rule 1: To multiply monomials with the same base, keep the base and add the powers:

  x ^ax ^b = x ^{a + b}

• Rule 2: To raise a base to a power, keep the base and multiply the powers.

  ( x ^a ) ^b = x ^ab

• Rule 3: To raise a product to a power, raise each factor in the product to that power.

  ( xy) ^a = x ^ay ^a

Example 1

Simplify each of the following multiplication problems and state which of the preceding rules was applied.

1. yy ⁵

2. ( x ⁴) ³

3. (–2 x ⁴ y ² z ³) ⁵

4. a ³( a ² b ³) ⁴

To multiply monomials together, follow this procedure.

1. Multiply the numerical coefficients together.

2. Multiply the variables together.

3. Write the results as a product.

Example 2

Simplify each of the following.

1. (4 x ²)(3 x ³)

2. (–8 a ³ b ²)(2 a ² b ²) ³

1. (4 x ²)(3 x ³) = (4 × 3)( x ² x ³) = 12 x ⁵

2. (–8 a ³ b ²)(2 a ² b ²) ³ = (–8 a ³ b ²)(8 a ⁶ b ⁶) = –64 a ⁹ b ⁸

To multiply polynomials together, multiply each term in one polynomial by each term in the other polynomial. Then simplify if possible.

Example 3

Multiply each of the following.

1. 5 x(3 x ² – 4 x + 2)

2. (4 x – 2)(3 x + 5)

3. ( x + y)( x ² – xy + y ²)

The following shows how each equation is multiplied both horizontally and vertically.

Equation (a) done horizontally:

Equation (a) done vertically:

Equation (b) done horizontally:

Equation (b) done vertically:

Equation (c) done horizontally:

Equation (c) done vertically: