Simplifying and Combining Radical Expressions Section 9.3 Simplifying and Combining Radical Expressions

Objectives Use the product rule to simplify radical expressions Use prime factorization to simplify radical expressions Use the quotient rule to simplify radical expressions Add and subtract radical expressions.

Objective 1: Use the Product Rule to Simplify Radical Expressions The Product Rule for Radicals: The nth root of the product of two numbers is equal to the product of their nth roots. If are real numbers, The product rule for radicals can be used to simplify radical expressions. When a radical expression is written in simplified form, each of the following is true. Each factor in the radicand is to a power that is less than the index of the radical. The radicand contains no fractions or negative numbers. No radicals appear in the denominator of a fraction. Read as “the nth root of a times b equals the nth root of a times the nth root of b.”

Objective 1: Use the Product Rule to Simplify Radical Expressions To simplify radical expressions, we must often factor the radicand using two natural number factors. To simplify square-root, cube-root, fourth-root radicals and so/on, it is helpful to memorize the following lists. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, . . . Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000, . . . Perfect-fourth powers: 1, 16, 81, 256, 625, . . . Perfect-fifth powers: 1, 32, 243, 1,024, . . .

EXAMPLE 1 Simplify : Strategy We will factor each radicand into two factors, one of which is a perfect square, perfect cube, or perfect-fourth power, depending on the index of the radical. Then we can use the product rule for radicals to simplify the expression. Why Factoring the radicand in this way leads to a square root, cube root, or fourth root of a perfect square, perfect cube, or perfect-fourth power that we can easily simplify.

EXAMPLE 1 Simplify : Solution a. To simplify , we first factor 12 so that one factor is the largest perfect square that divides 12. Since 4 is the largest perfect-square factor of 12, we write 12 as 4  3, use the product rule for radicals, and simplify.

EXAMPLE 1 Simplify : Solution b. The largest perfect-square factor of 98 is 49. Thus,

EXAMPLE 1 Simplify : Solution c. Since the largest perfect-cube factor of 54 is 27, we have Write 54 as 27  2. The cube root of a product is equal to the product of the cube roots: Multiply: 5  3 = 15.

EXAMPLE 1 Simplify : Solution d. The largest perfect-fourth power factor of 48 is 16. Thus,

Objective 2: Use Prime Factorization to Simplify Radical Expressions When simplifying radical expressions, prime factorization can be helpful in determining how to factor the radicand.

Simplify. All variables represent positive real numbers. EXAMPLE 3 Strategy In each case, the way to factor the radicand is not obvious. Another approach is to prime-factor the coefficient of the radicand and look for groups of like factors. Why Identifying groups of like factors of the radicand leads to a factorization of the radicand that can be easily simplified.

Simplify. All variables represent positive real numbers. EXAMPLE 3 Solution

Simplify. All variables represent positive real numbers. EXAMPLE 3 Solution

Simplify. All variables represent positive real numbers. EXAMPLE 3 Solution

Objective 3: Use the Quotient Rule to Simplify Radical Expressions The Quotient Rule for Radicals: The nth root of the quotient of two numbers is equal to the quotient of their nth roots. Read as “the nth root of a divided by b equals the nth root of a divided by the nth root of b.”

Simplify each expression: (All variables represent positive real numbers.) EXAMPLE 4 Strategy In each case, the radical is not in simplified form because the radicand contains a fraction. To write each of these expressions in simplified form, we will use the quotient rule for radicals. Why Writing these expressions in form leads to square roots of perfect squares and cube roots of perfect cubes that we can easily simplify.

Simplify each expression: (All variables represent positive real numbers.) EXAMPLE 4 Solution a. We can use the quotient rule for radicals to simplify each expression.

Simplify each expression: (All variables represent positive real numbers.) EXAMPLE 4 Solution

Objective 4: Add and Subtract Radical Expressions Radical expressions with the same index and the same radicand are called like or similar radicals. For example, are like radicals. are not like radicals, because the radicands are different. are not like radicals, because the indices are different. To add or subtract radicals, simplify each radical, if possible, and combine like radicals.

EXAMPLE 6 Simplify: Strategy Since the radicals in each part are unlike radicals, we cannot add or subtract them in their current form. However, we will simplify the radicals and hope that like radicals result. Why Like radicals can be combined.

EXAMPLE 6 Solution a. We begin by simplifying each radical expression:

EXAMPLE 6 Solution b. We begin by simplifying each radical expression: