Chapter 7 Radical Expressions Solving Equations

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7-5 Operations with Radical Expressions:

7-5 Operations with Radical Expressions

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Definition nth Root For any real numbers a and b, any positive integer n, if a n = b, then a is an nth root of b. Since 5 2 = 25, 5 is a square root of 25. Since 5 3 = 125, 5 is a cube root of 125. Since 5 4 = 625, 5 is a fourth root of 625. Since 5 5 = 3125, 5 is a fifth root of 3125. This pattern leads to the definition of nth root.

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A radical sign is used to indicate a root. The number under the radical sign is the radicand . The index gives the degree of the root. radical sign

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When a number has two real roots, the positive root is called the principal root and the radical sign indicates the principal root. The principal fourth root of 16 is written The principal fourth root of 16 is 2 because equals . The other fourth root of 16 is written as which equals -2. √16 4 √16 √2 4 4 4 - √16 4

Example 2 Finding Roots:

Example 2 Finding Roots Find each real number root. √-27 √81 √49 3 4

Example 3a Simplifying Radical Expressions:

Example 3a Simplifying Radical Expressions Simplify each radical expression. √4x 6 √a 3 b 6 √x 4 y 8 3 4

Example 3b Simplifying Radical Expressions:

Example 3b Simplifying Radical Expressions Simplify each radical expression. √4x 2 y 4 √-27c 6 √x 8 y 12 3 4

nth Roots:

Simplify the following. n th Roots Example

7.4:

7.4 Simplifying Radicals

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If and are real numbers, Product Rule for Radicals

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Simplify the following radical expressions. No perfect square factor, so the radical is already simplified. Simplifying Radicals Example

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Simplify the following radical expressions. Simplifying Radicals Example

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If and are real numbers, Quotient Rule for Radicals

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Simplify the following radical expressions. Simplifying Radicals Example

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Adding and Subtracting Radicals

Sums and Differences:

Sums and Differences Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences.

Like Radicals:

In previous chapters, we’ve discussed the concept of “like” terms. These are terms with the same variables raised to the same powers. They can be combined through addition and subtraction. Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand. Like radicals are radicals with the same index and the same radicand. Like radicals can also be combined with addition or subtraction by using the distributive property. Like Radicals

Adding and Subtracting Radical Expressions:

Can not simplify Can not simplify Adding and Subtracting Radical Expressions Example

Adding and Subtracting Radical Expressions:

Simplify the following radical expression. Example Adding and Subtracting Radical Expressions

Adding and Subtracting Radical Expressions:

Simplify the following radical expression. Example Adding and Subtracting Radical Expressions

Adding and Subtracting Radical Expressions:

Simplify the following radical expression. Assume that variables represent positive real numbers. Example Adding and Subtracting Radical Expressions

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Multiplying and Dividing Radicals

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If and are real numbers, Multiplying and Dividing Radical Expressions

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Simplify the following radical expressions. Multiplying and Dividing Radical Expressions Example

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Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator. If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator . This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator. Rationalizing the Denominator

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Rationalize the denominator. Rationalizing the Denominator Example

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Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical. In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing). The conjugate uses the same terms, but the opposite operation (+ or  ). Conjugates

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Rationalize the denominator. Rationalizing the Denominator Example

PRACTICE:

PRACTICE Multiply. Simplify if possible. √3 √12 √3 √-9 √4 √ -4 3 4 3 4 * * *

PRACTICE:

PRACTICE Simplify each expressions. Assume that all variables are positive. √50x 4 √18x 4 3√7x 3 2√21x 3 y 2 * 3

PRACTICE:

PRACTICE Multiply and simplify. 3√7x 3 2√21x 3 y 2 √54x 2 y 3 √5x 3 y 4 * 3 3 *

PRACTICE:

PRACTICE Multiply. Simplify if possible. √243 √12x 4 √27 √3x √1024x 15 √4x

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To rationalize a denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.

PRACTICE:

PRACTICE Rationalize the denominator of each expression. 7 √2x 3 √4 5 √10xy √6x

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Like radicals are radical expressions that have the same index and the same radicand. To add or subtract like radicals, use the Distributive Property.

PRACTICE:

PRACTICE 5 √ x - 3 √ x 4 √ xy + 5 √ xy 4 √ 2 - 5 √ 3 7 √ 5 - 2 √5 2 √ 7 + 3 √ 7 3 3 4 3

PRACTICE:

PRACTICE 6 √ 18 + 4 √ 8 - 3√ 72 √ 50 + 3 √ 32 - 5 √ 18

PRACTICE:

PRACTICE (3 + 2 √ 5 ) ( 2 + 4 √ 5 ) ( √ 2 - √ 5 ) 2

PRACTICE:

PRACTICE (2 + √ 3 ) ( 2 - √ 3 ) ( √ 2 - √ 5 ) ( √ 2 + √ 5 )

PRACTICE Rationalizing a Binomial Radical Denominator :

PRACTICE Rationalizing a Binomial Radical Denominator 3 + √5 1 - √5 6 + √15 4 - √15

7-4 Rational Exponents:

7-4 Rational Exponents What you’ll learn … To simplify expressions with rational exponents 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

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Another way to write a radical expression is to use a rational exponent. Like the radical form, the exponent form always indicates the principal root. √25 = 25 ½ √27 = 27 ⅓ 3 √16 = 16 1/4 4

Example 1 Simplifying Expressions with Rational Exponents:

Example 1 Simplifying Expressions with Rational Exponents 125 1/3 5 ½ 2 ½ 2 ½ 2 ½ 8 ½ * * P/R = power/root √ x p r ( √x ) p r

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A rational exponent may have a numerator other than 1. The property (a m ) n = a mn shows how to rewrite an expression with an exponent that is an improper fraction. Example 25 3/2 = 25 (3*1/2) = (25 3 ) ½ = √25 3

Example 2 Converting to and from Radical Form:

Example 2 Converting to and from Radical Form x 3/5 y -2.5 y -3/8 √a 3 ( √b ) 2 √x 2 5 3

Properties of Rational Exponents:

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example a m * a n = a m+n 8 ⅓ * 8 ⅔ = 8 ⅓ + ⅔ = 8 1 =8 (a m ) n = a mn (5 ½ ) 4 = 5 ½ * 4 = 5 2 = 25 (ab) m = a m b m (4 *5) ½ = 4 ½ * 5 ½ =2 * 5 ½

Properties of Rational Exponents:

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example a -m 1 9 - ½ 1 1 a m 9 ½ 3 a m a m-n π 3/2 π 3/2-1/2 = π 1 = π an π ½ a m a m 5 5 ⅓ 5 ⅓ b b m 27 27 ⅓ 3 = = = = = = ⅓ =

Example 4 Simplifying Numbers with Rational Exponents:

Example 4 Simplifying Numbers with Rational Exponents (-32) 3/5 4 -3.5

Example 5 Writing Expressions in Simplest Form:

Example 5 Writing Expressions in Simplest Form (16y -8 ) -3/4 (8x 15 ) -1/3

7-7 Solving Radical Equations:

7-7 Solving Radical Equations What you’ll learn … To solve radical equations

Example 2 Solving Radical Equations with Rational Exponents:

Example 2 Solving Radical Equations with Rational Exponents Solve 2 (x – 2) 2/3 = 50 3(x+1) 3/5 = 24

Example 4 Checking for Extraneous Solutions:

Example 4 Checking for Extraneous Solutions Solve √x – 3 + 5 = x √3x + 2 - √2x + 7 = 0

Example 5 Solving Equations with Two Rational Exponents:

Example 5 Solving Equations with Two Rational Exponents Solve (2x +1) 0.5 – (3x+4) 0.25 = 0 Solve (x +1) 2/3 – (9x+1) 1/3 = 0

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