§ 7.4 Adding, Subtracting, and Dividing Radical Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.4 Combining RadicalsEXAMPLE Simplify (add or subtract) by combining like radical terms: SOLUTION Apply the distributive property. Simplify. Apply the distributive property. Group like terms. Simplify.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.4 Simplifying Radicals The Quotient Rule for Radicals If and are real numbers and, then The nth root of a quotient is the quotient of the nth roots.

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.4 Simplifying RadicalsEXAMPLE Simplify using the quotient rule: SOLUTION

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.4 Combining Radicals Dividing Radical Expressions If and are real numbers and, then To divide two radical expressions with the same index, divide the radicands and retain the common index.

DONE

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.4 Combining RadicalsEXAMPLE Divide and, if possible, simplify: SOLUTION In each part of this problem, the indices in the numerator and the denominator are the same. Perform each division by dividing the radicands and retaining the common index. Divide the radicands and retain the common index. Divide factors in the radicand. Subtract exponents on common bases.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.4 Combining RadicalsCONTINUED Simplify. Factor using the greatest perfect square factor. Factor into two radicals. Simplify. Divide the radicands and retain the common index. Divide factors in the radicand. Subtract exponents on common bases.

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.4 Combining RadicalsCONTINUED Simplify. Factor using the greatest perfect square factor. Factor into two radicals. Simplify.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.4 Combining Radicals Important to remember: Like radicals have the same indices and radicands. Like radicals can be added or subtracted using the distributive property. In some cases, you cannot see that radicals are “like” until you simplify them. When attempting to combine radicals, you should simplify the radicals first. Then you may see that you have like radicals that can be combined. Are we like? You don’t look like me. Yep. I’m 2 square roots of 3 and you are 5 square roots of 3. We have the same indices and radicands. We’re like! Let’s see…2 of them + 5 of them = 7 of them

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.4 Combining Radicals Apples to apples, oranges to oranges,… you can only add “like” things…. Two or more radical expressions that have the same indices and the same radicands are called like radicals. Like radicals can be combined under addition in exactly the same way that we combined like terms under addition. Examples of this process follow. 2 elephants + 3 elephants = 5 elephants but 5 tigers + 3 gorillas = ???

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.4 Combining RadicalsEXAMPLE Simplify by combining like radical terms, if possible: SOLUTION Factor the radicands using the greatest perfect square factors. Apply the distributive property. Take the square root of each factor.

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.4 Combining Radicals Factor the radicands using the greatest perfect cube factors. Simplify. Take the cube root of each factor. CONTINUED Apply the distributive property. Simplify.