§ 7.5 Multiplying With More Than One Term and Rationalizing Denominators

Blitzer, Intermediate Algebra, 4e – Slide #63 Multiplying RadicalsEXAMPLE Multiply: SOLUTION Use the distributive property. Multiply the radicals. Use FOIL. Multiply the radicals.

Blitzer, Intermediate Algebra, 4e – Slide #64 Multiplying Radicals Factor the third radicand using the greatest perfect square factor. CONTINUED Factor the third radicand into two radicals. Simplify.

Blitzer, Intermediate Algebra, 4e – Slide #65 Multiplying RadicalsEXAMPLE Multiply: SOLUTION Group like terms. Multiply the radicals. Use FOIL. Combine radicals.

Blitzer, Intermediate Algebra, 4e – Slide #66 Multiplying Radicals Multiply the radicals. Use the special product for CONTINUED

Blitzer, Intermediate Algebra, 4e – Slide #67 Rationalizing DenominatorsEXAMPLE Rationalize each denominator: SOLUTION (a) Using the quotient rule, we can express. We have cube roots, so we want the denominator’s radicand to be a perfect cube. Right now, the denominator’s radicand is. We know that If we multiply the numerator and the denominator of, the denominator becomes

Blitzer, Intermediate Algebra, 4e – Slide #68 Rationalizing Denominators The denominator no longer contains a radical. Therefore, we multiply by 1, choosing CONTINUED Use the quotient rule and rewrite as the quotient of radicals. Multiply the numerator and denominator by to remove the radical in the denominator. Multiply numerators and denominators.

Blitzer, Intermediate Algebra, 4e – Slide #69 Rationalizing DenominatorsCONTINUED Simplify. (b) The denominator, is a fifth root. So we want the denominator’s radicand to be a perfect fifth power. Right now, the denominator’s radicand is We know that If we multiply the numerator and the denominator of, the denominator becomes

Blitzer, Intermediate Algebra, 4e – Slide #70 Rationalizing DenominatorsCONTINUED The denominator’s radicand is a perfect 5 th power. The denominator no longer contains a radical. Therefore, we multiply by 1, choosing Write the denominator’s radicand as an exponential expression. Multiply the numerator and the denominator by Multiply the numerators and denominators.

Blitzer, Intermediate Algebra, 4e – Slide #71 Rationalizing DenominatorsCONTINUED Simplify.

Blitzer, Intermediate Algebra, 4e – Slide #72 Rationalizing DenominatorsEXAMPLE Rationalize each denominator: SOLUTION (a) The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we multiply by 1, choosing Multiply by 1.

Blitzer, Intermediate Algebra, 4e – Slide #73 Rationalizing DenominatorsCONTINUED Evaluate the exponents. Subtract. Divide the numerator and denominator by Multiply by 1.

Blitzer, Intermediate Algebra, 4e – Slide #74 Rationalizing DenominatorsCONTINUED Simplify. (b) The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we multiply by 1, choosing Multiply by 1.

Blitzer, Intermediate Algebra, 4e – Slide #75 Rationalizing DenominatorsCONTINUED Multiply by 1. Rearrange terms in the second numerator. Simplify.

Blitzer, Intermediate Algebra, 4e – Slide #76 Rationalizing NumeratorsEXAMPLE Rationalize the numerator: SOLUTION The conjugate of the numerator is If we multiply the numerator and the denominator by the simplified numerator will not contain a radical. Therefore, we multiply by 1, choosing Multiply by 1.

Blitzer, Intermediate Algebra, 4e – Slide #77 Rationalizing Numerators Leave the denominator in factored form. CONTINUED Evaluate the exponents. Simplify the numerator. Simplify by dividing the numerator and denominator by 7.