§ 7.3 Multiplying and Simplifying Radical Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.3 Multiplying Radicals The Product Rule for Radicals If and are real numbers, then The product of two nth roots is the nth root of the product.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.3 Multiplying RadicalsEXAMPLE Multiply: SOLUTION In each problem, the indices are the same. Thus, we multiply the radicals by multiplying the radicands.

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.3 Simplifying Radicals Simplifying Radical Expressions by Factoring A radical expression whose index is n is simplified when its radicand has no factors that are perfect nth powers. To simplify, use the following procedure: 1) Write the radicand as the product of two factors, one of which is the greatest perfect nth power. 2) Use the product rule to take the nth root of each factor. 3) Find the nth root of the perfect nth power.

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.3 Simplifying RadicalsEXAMPLE Simplify by factoring: SOLUTION 4 is the greatest perfect square that is a factor of 28. Take the square root of each factor. Write as 2. is the greatest perfect cube that is a factor of the radicand.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.3 Simplifying Radicals Factor into two radicals. CONTINUED Take the cube root of

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.3 Simplifying RadicalsEXAMPLE If, express the function, f, in simplified form. SOLUTION Begin by factoring the radicand. There is no GCF. This is the given function. Factor 48. Rewrite 8 as. Take the cube root of each factor. Take the cube root of and

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.3 Simplifying Radicals Simplifying When Variables to Even Powers in a Radicand are Nonnegative Quantities For any nonnegative real number a,

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.3 Simplifying RadicalsEXAMPLE Simplify: SOLUTION We write the radicand as the product of the greatest perfect square factor and another factor. Because the index of the radical is 2, variables that have exponents that are divisible by 2 are part of the perfect square factor. We use the greatest exponents that are divisible by 2. Use the greatest even power of each variable. Group the perfect square factors. Factor into two radicals.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.3 Simplifying RadicalsCONTINUED Simplify the first radical. Remember – we take the principal root which is the positive root if there is a choice between a negative and a positive. Odd roots of a negative number are negative and we don’t have a choice. When we take the even root of a positive number – we could argue that we might get either a negative or a positive. But we take the positive, for that is the principal root. Since we don’t know the sign of the variable x in this problem and x could be either negative or positive – we just put absolute value bars around the x to assure that we have chosen the positive root.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.3 Simplifying RadicalsEXAMPLE Simplify: SOLUTION We write the radicand as the product of the greatest 4 th power and another factor. Because the index is 4, variables that have exponents that are divisible by 4 are part of the perfect 4 th factor. We use the greatest exponents that are divisible by 4. Identify perfect 4 th factors. Group the perfect 4 th factors. Factor into two radicals. Simplify the first radical.

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.3 Multiplying RadicalsEXAMPLE Multiply and simplify: SOLUTION Use the product rule. Multiply. Identify perfect 4 th factors. Group the perfect 4 th factors. Factor into two radicals.

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.3 Multiplying Radicals Use the product rule. Multiply. Identify perfect 5 th factors. Group the perfect 5 th factors. Factor into two radicals. CONTINUED Simplify the first radical. Factor into two radicals.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.3 Simplifying Radicals Important to Remember: A radical expression of index n is not simplified if you can take any roots – that is if there are any factors of the radicand that are perfect nth powers. Take all roots.