§ 7.3 Multiplying and Simplifying Radical Expressions

The Product Rule for Radicals 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 #2 Section 7.3

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

Multiplying Radicals Multiply: Check Point 1 on p 509 Multiply: indices are the same. Note: problems from 1-19. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.3

Simplifying Radicals Simplify by factoring: EXAMPLE 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. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.3

Simplifying Radicals CONTINUED is the greatest perfect cube that is a factor of the radicand. Factor into two radicals. Take the cube root of Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.3

Simplifying Radicals Check Point 2 on p 510 problems from 21-31 Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.3

Simplifying Radicals Simplify: EXAMPLE 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. Simplify the first radical. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.3

Simplifying Radicals Simplify: EXAMPLE Simplify: SOLUTION We write the radicand as the product of the greatest 4th power and another factor. Because the index is 4, variables that have exponents that are divisible by 4 are part of the perfect 4th factor. We use the greatest exponents that are divisible by 4. Identify perfect 4th factors. Group the perfect 4th factors. Factor into two radicals. Simplify the first radical. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.3

Simplifying Radicals Check Point 4 and 5 on p 512 problems from 39-47 Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.3

Simplifying Radicals Check Point 7 on p 513 problems from 61-77 Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.3

DONE

Simplifying Radical Expressions by Factoring 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 #13 Section 7.3

Simplifying Radicals If , express the function, f, in simplified form. EXAMPLE 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 #14 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 #15 Section 7.3

Multiplying Radicals Multiply and simplify: Use the product rule. EXAMPLE Multiply and simplify: SOLUTION Use the product rule. Multiply. Identify perfect 4th factors. Group the perfect 4th factors. Factor into two radicals. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.3

Multiplying Radicals Factor into two radicals. Use the product rule. CONTINUED Factor into two radicals. Use the product rule. Multiply. Identify perfect 5th factors. Group the perfect 5th factors. Factor into two radicals. Simplify the first radical. Blitzer, Intermediate Algebra, 5e – Slide #17 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. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.3