5.0 Perfect squares and Perfect Cubes
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2 5.0 Perfect squares and Perfect Cubes A fast and efficient way to solve radicals is to recognize and know the perfect numbers. Perfect Squares Perfect Cubes Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 9
3 5.1a Working with Radical Numbers Definition: Radical Index Radicand n x Recap: Exponent Laws ( a b) a b x x Extension of Exponent Laws: a a b b In General: n a b = n a n b a n a n = b n b Recall: Square roots a number r, is a square root of a number x, if r x. Note: A positive number always has two square roots, one positive and one negative, because: It is impossible to obtain a negative number when a number is squared, therefore the square root of a negative number is NOT defined. Notation: radical sign, denotes the positive square root only. x means the positive square root of x, where x 0. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page of 9
4 Example 1: Simplify. a) 500 = b) 0.16 = Cube Roots A number r, is a cube root of a number x, if r x. Note: The cube root of a positive number is. The cube root of a negative number is. x means the cube root of x Example : Simplify. Round to decimal places if necessary. a) 64 b) 15 c ) 1 b) 15 note: is the same as Higher Roots An expression of the form n x is a radical, where n is a natural number. If n is even, the expression represents only the positive root. Example : Simplify. a) 4 81 b) 5 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page of 9
5 Entire Radical vs. Mixed Radical Let s look examine the radical : can be expressed as a mixed radical different Methods Method A (Prime Factorization) Method B (Product of radicals) Example 4: Express each entire radical as a mixed radical (use method B) a) b) c) d) Example 5: Express as an entire radical. a) b) Example 6: Without a calculator order the set of numbers from least to greatest. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 4 of 9
6 In all the previous examples so far, the radicand has been a constant term. But what if the radicand contained a variable? Since a variable is representative of a number, we need to ask ourselves if there are any restrictions to what this number can be given the index of the radical. Let s take a look at the radical expression What are the values for which this expression is defined? Let us now look at the expression What are the values for which this expression is defined? Can the expression be simplified? Example 7: Determine the variable for which the expressions are defined. Simplify the expressions. a) b) c) Assignment Page 78 (min) #1, ad, 4, 6bc, 14, (worksheet) 6 (reg) #1 4, 6, 14, (worksheet) 6 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 5 of 9
7 5.1a Homework Worksheet 6. i. Identify the variables for which the expression is defined ii. Simplify the expression a) b) c) Ans: 6a) i. ii. b) i. ii. c) i. ii. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 6 of 9
8 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 7 of 9
9 5.1b Adding and Subtracting Radicals Similar to adding and subtracting polynomials or fractions, radicals have their own set of rules to simplify radicals. Recall: 1) + 4 = ) Think: ) You are only allowed to add/subtract polynomials that have like terms. Similarly when you are add/subtract radicals, you can only do so when you have like radicals. Like radicals are radicals that have the same index and same radicand. Similar to the like terms such as question ) (where we add the coefficients), we add the numbers that are in front of the radical signs. Example 1: Simplify. a) b) c) In cases where the index and radicand are not the same, we need to simplify the radicals first in order to see if we have any like radicals to add or subtract. Example : Simplify. a) b) c) d) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 8 of 9
10 Some radical expressions contain variables. Before you attempt to add/subtract the expressions, you must state the values for which the expressions are defined. Example : State the values for which the expression is defined, then simplify. a) b) c) Example 4: State the values for which the expression is defined, then simplify. a) 4 c 4 9c b) 0x 45x c) 16 54x x d) 0x 80x Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 9 of 9
11 Example 5: A square is inscribed in a circle. The area of the circle is 40 m. a) What is the exact length of the diagonal of the square? b) Determine the exact perimeter of the square. Assignment Page 78 (min) #5cd, 8cd, 9bd, 10bd, 15 (reg) #5, 8 10, 15, 19 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 10 of 9
12 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 11 of 9
13 5. Multiplying and Dividing Radical Expressions Multiplying Radical Expressions The rule of multiplying radicals: a b ab, a 0, b 0 Note: The expression 4 5 has the same meaning as 4 5 When multiplying radicals the index must be the same. You will calculate the product of the values outside the radical sign and the product of the values inside the radical sign. Example 1: Simplify a) b) c) x 4 6 When multiplying radical expressions that contain more than one term, we will use the distributive property or F.O.I.L in order to get rid of the brackets. Example : Expand and simplify a) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 9
14 Example : Expand and simplify. State an y restrictions for the variables. a) b) Dividing Radical Expressions The rule for dividing radicals: a b a b when, You can combine two radicals into one or separate 1 radical into separate ones. Example 4: Simplify. a) 0 b) c) 9x x IMPORTANT: Radicals must NEVER be left in the denominator. If there is a radical in the denominator, the method to get rid of them is called to RATIONALIZE THE DENOMINATOR Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 9
15 Given the expression: 1 If we multiply the numerator and denominator by the radical in the denominator,. The radical will move to the numerator and the answer will remain the same. 1 Example 5: Rationalize the denominator then simplify. a) b) c) Note: Simplify first, it prevents you from doing more work! Conjugate of a Binomial Expression: The conjugate of is ( ) When the denominator of an expression contains two terms with at least one radical sign, in order to rationalize the denominator, you must multiply both the numerator and denominator by the conjugate of the denominator. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 14 of 9
16 Example 6: Rationalize the denominator. Simplify a) b) Assignment Page 89, (min) #1be, ad, bc, 4ad, 5c, 6d, 8c, 10d, 11d, 17 (reg) #1bdef, ad, bc, 4ade, 5bcd, 6d, 7a, 8c, 10bcd, 11cd, 17 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 15 of 9
17 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 16 of 9
18 Mid-Unit Activity 1. The area of a rectangle is 16 square units. If the width of the rectangle is units, determine the exact value of the perimeter in simplified radical form with a rationalized denominator. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 17 of 9
19 . Determine the exact value of both the area and perimeter of the following isosceles triangle. Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 18 of 9
20 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 19 of 9
21 5. Radical Equations Radical Equation: When solving a radical equation we must first state the restrictions on the variable. Because we are taking the square root, there are certain values for the variable that will make the radical expression undefined. To find the restrictions, knowing that the radicand must be a positive number or zero (radicand, set up an inequality and solve for the variable Extraneous Roots: Solution(s) that may arise when an original equation is altered in order to solve the equation. Any roots that are not part of the domain of the original equation are called extraneous roots. To solve a radical equation algebraically: Step 1: Isolate the radical on one side of the equation. If there are two, isolate the most complex term. Step : Square each side, then solve the equation that results. (note: If the resulting equation still contains a radical term, repeat steps 1 and ) Step : Identify extraneous roots and reject them. Example 1: State the restrictions for, them solve for x. a) b) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 0 of 9
22 Example : State the restrictions for, then solve for x. a) x b) x 1 x Example : State the restrictions for, then solve the equation. x x 5 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 9
23 Example 4: The period, T, seconds, of a pendulum is related to its length, L, in metres. The period is L the time to complete one full cycle and can be approximated with the formula T. 10 a) Write an equivalent formula with a rational denominator. b) The length of the pendulum in the HSBC building in downtown Vancouver is 7 m. How long would the pendulum take to complete cycles (nearest tenth). Example 5: The mass, m, in kilograms, that a beam with a fixed width and length can support is t 1 m 5 related to its thickness,, in cetimetres. The formula is t, m 0. If a beam is 4 cm thick, what mass can it support? Assignment (Min) Page 91, #19ab Page 01, #4bd, 6c, 7d, 8d, 9cd, 10c, 14, 16,18 (Reg) Page 91, #19ab Page 01, #4bd, 5, 6bc, 7cd, 8ad, 9bcd, 10bcd, 14, 16,18 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page of 9
24 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page of 9
25 Ch. 5 Review Key Ideas Restrictions and properties of radical expressions Simplifying radical expressions Adding & subtracting radical expressions Multiplying & Dividing radical expressions Rationalizing the denominator Solving Radical equations Example 1: Determine the values of the variables for which the expres sion is defined. a) a) b) c) Example : Simplify the following expressions. a) b) c) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 4 of 9
26 Example : Simplify the following expressions. a) b) c) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 5 of 9
27 Example 4: Rationalize the denominator. a) b) c) d) Example 5: Determine the values for which the expressions are defined, then solve. a) Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 6 of 9
28 Example 5: Determine the values for which the expressions are defined, then solve. b) c) d) Assignment Page 04, #1ad, cd, c, 4ab, 6 (nc), 8, 10bc, 11bc, 1ac, 14 17, 18bd, 19ade, 1 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 7 of 9
29 Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 8 of 9
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