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Chapter 15 Roots and Radicals

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**Introduction to Radicals**

15.1 Introduction to Radicals

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**Radicands The symbol is called a radical or radical sign.**

The expression under a radical sign is the radicand. A radical expression is an expression containing a radical sign.

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**Square Root If a is a positive number, then**

is the positive square root of a and is the negative square root of a. Also, = 0. Note: A square root of a negative number is not a real number.

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Square Roots The reverse operation of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a number that, when squared, equals a.

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Perfect Squares Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. They cannot be written as a quotient of integers. If needed, you can find a decimal approximation for these irrational numbers on a calculator. Otherwise, leave them in radical form.

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**Finding Square Roots Example Find each square root. Approximate**

to three decimal places. ≈ ≈ 2.236

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**Cube Root The cube root of a real number a**

Note: a is not restricted to non-negative numbers for cubes.

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Finding Cube Roots Example Find each cube root. = 3 = –5

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**Finding nth Roots Other roots can be found, as well.**

The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number.

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**Finding nth Roots Example Find each root. 2 –2 = =**

is not a real number since the index, 4, is even and the radicand, –16, is negative. There is no real number that when raised to the 4th power gives –16.

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**Simplifying Radicals Containing Variables**

Radicands might also contain variables and powers of variables. To make sure that simplifies to a nonnegative number, we have the following. For any real number a Now, if x is a negative number, like x = –2, then x2 = = = 2, not –2, our original x. x2 a2 = |a| For example, = |–9| = 9 .

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**Simplifying Radicals Containing Variables**

To avoid this confusion, for the rest of the chapter we assume that if a variable appears in the radicand of a radical expression, it represents positive numbers only.

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**Simplifying Radicals Containing Variables**

Example Find each root.

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15.2 Simplifying Radicals

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**Product Rule for Radicals**

If and are real numbers, then

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**Quotient Rule for Radicals**

If and are real numbers and b ≠ 0, then

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**Simplifying Radicals Simplify the following radical expressions.**

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

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**Simplifying Radicals Simplify the following radical expressions.**

Example Simplify the following radical expressions.

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**Simplifying Radicals Simplify the following radical expressions.**

Example Simplify the following radical expressions.

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**Adding and Subtracting Radicals**

15.3 Adding and Subtracting Radicals

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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.

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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.

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**Adding and Subtracting Radical Expressions**

Example Can not simplify Can not simplify

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**Adding and Subtracting Radical Expressions**

Example Simplify the following radical expression.

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**Adding and Subtracting Radical Expressions**

Example Simplify the following radical expression.

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**Adding and Subtracting Radical Expressions**

Example Simplify the following radical expression. Assume that variables represent positive real numbers.

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**Multiplying and Dividing Radicals**

15.4 Multiplying and Dividing Radicals

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**Product Rule for Radicals**

If and are real numbers, then

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**Multiplying and Dividing Radical Expressions**

Example Simplify the following radical expression.

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**Quotient Rule for Radicals**

If and are real numbers and b ≠ 0, then

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**Multiplying and Dividing Radical Expressions**

Example Simplify the following radical expression.

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**Rationalizing Denominators**

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.

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**Rationalizing the Denominators**

Example Rationalize the denominator.

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Conjugates 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 –).

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**Rationalizing the Denominator**

Example Rationalize the denominator.

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**Solving Equations Containing Radicals**

15.5 Solving Equations Containing Radicals

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**The Squaring Property of Equality**

If a = b, then a2 = b2.

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Extraneous Solutions If both sides of an equation are squared, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution. For this reason, we must always check the proposed solutions of radical equations in the original equation.

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Helpful Hint Don’t forget to check the proposed solutions of radical equations in the original equation.

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**Solving Radical Equations**

Example Solve the following radical equation. Check: Substitute into the original equation. True So the solution is x = 24.

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**Solving Radical Equations**

Example Solve the following radical equation. Check: Substitute into the original equation. Does NOT check, since the left side of the equation is asking for the principal square root. So this equation has no solution.

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**Solving Radical Equations**

To Solve a Radical Equation Containing Square Roots Step 1: Arrange terms so that one radical is by itself on one side of the equation. That is, isolate a radical. Step 2: Square both sides of the equation. Step 3: Simplify both sides of the equation. Continued

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**Solving Radical Equations**

To Solve a Radical Equation Containing Square Roots (cont’d) Step 4: If equation still contains a radical, repeat Steps 1 through 3. Step 5: Solve the equation. Step 6: Check all solutions in the original equation for extraneous solutions.

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**Solving Radical Equations**

Example Solve the following radical equation. Check: Substitute into the original equation. True So the solution is x = 2.

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**Solving Radical Equations**

Example Solve the following radical equation. Continued

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**Solving Radical Equations**

Example continued Substitute the value for x into the original equation, to check the solution. True 21/4 is an extraneous solution. The only solution is x = 3. False

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**Solving Radical Equations**

Example Solve the following radical equation. Continued

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**Solving Radical Equations**

Example continued Substitute the value for x into the original equation, to check the solution. This equation has no solution. False

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**Solving Radical Equations**

Example Solve the following radical equation. Continued

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**Solving Radical Equations**

Example continued Substitute the value for x into the original equation, to check the solution. True True The solutions are x = 4 or x = 20.

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**Radical Equations and Problem Solving**

15.6 Radical Equations and Problem Solving

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**The Pythagorean Theorem**

If a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 + b2 = c2.

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**Using the Pythagorean Theorem**

Example Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches. c2 = = = 53 c = inches

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