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## Chapter 5

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**Chapter 5**Radical Expressions and Equations**5.1 – Working with radicals**Chapter 5: Radical Expressions & Equations**radicals**radical radicand index (is 2) Pairs of Like Radicals: Pairs of Unike Radicals: Mixed Radical: Entire Radical:**example**Express each mixed mixed radical in entire radical form. Identify the values of the variable for which the radical represents a real number. a) b) c) a) b) c) Try it!**Radicals in simplest form**• A radical is in simplest form if the following are true: • The radicand does not contain a fraction or any factor which could be removed. • The radical is not part of the denominator of a fraction. Not the simplest form, since 18 has 9 as a factor, which is a perfect square. Simplest form!**The Rules**• You can take a number out from under a root sign by square-rooting (or cube-rooting, if it’s a cubed root) the number. • Ex: • You can bring a number inside the a root sign by squaring (or cubing, if it’s a cubed root) the number. • Ex: • You can add or subtract like radicals. • Ex: • You cannot add or subtract unlike radicals.**example**Convert each entire radical to a mixed radical in simplest form. a) b) c) b) a) Find a factor that is a perfect square! c) Try it!**example**Simplify radicals and combine like terms.**P. 278-281 # 4, 6, 9, 10, 11, 13, 15, 17, 19, 21**Independent Practice**5.2 – Multiplying and dividing radical expressions**Chapter 5: Radical Expressions & Equations**Multiplying radicals**When multiplying radicals, multiply the coefficients and multiply the radicands. You can only multiply radicals if they have the same index. In general, , where k is a natural number, and m, n, a, and b are real numbers. If k is even, then a ≥ 0 and b ≥ 0.**example**Multiply. Simplify the products where possible. a) b) a) b) Try it!**Dividing radicals**The rules for dividing radicals are the same as the rules for multiplying them. In general, , where k is a natural number, and m, n, a, and b are real numbers. n ≠ 0 and b ≠ 0. If k is even, then a ≥ 0 and b> 0.**Rationalizing denominators**To simplify an expression that has a radical in the denominator, you need to rationalize the denominator. To rationalize means to convert to a rational number without changing the value of the expression. Example: When there is a monomial square-root denominator, we can rationalize the expression by multiplying both the numerator and the denominator by the same radical.**conjugates**For a binomial denominator that contains a square root, we must multiply both the numerator and denominator by a conjugate of the denominator. Conjugates are two binomial factors whose product is the difference of two squares. • the binomials (a + b) and (a – b) are conjugates since their product is a2 – b2 • Similarly, • So, they are conjugates! • The conjugate of is**examples**Simplify each expression. a) b) b) a) Try it!**Challenge: try it!**Simplify:**PG. 289-293, #4, 5, 10, 11, 13, 15, 17, 20, 21, 22, 23, 24,**26 Independent Practice**5.3 – Radical equations**Chapter 5: Radical Expressions & Equations**handout**You will need three metre sticks for the activity. Answer all the questions to the best of your abilities, as this is a summative assessment. You will need to use your prior knowledge of radical expressions, as well as your understanding of triangles.**Solving radical equations**• When solving a radical equation, remember to: • identify any restrictions on the variable • identify whether any roots are extraneous by determining whether the values satisfy the original equation.**example**a) State the restrictions on x in if the radical is a real number. b) Solve • You can’t take the square root of a negative number, so: • 2x – 1 ≥ 0 • 2x ≥ 1 • x ≥ ½ • So, the equation has a real solution when x ≥ ½. b) x = 25 meets the restrictions, and we can substitute it into the equation to check that it’s a solution: So, x = 25 is the solution.**example**Solve We need to check that they work:**example**What is the speed, in metres per second, of a 0.4-kg football that has a 28.8 J of kinetic energy? Use the kinetic energy formula, Ek = ½mv2, where Ek represents the kinetic energy in joules; m represents mass, in kilograms; and v represents speed, in metres per second. Substitute in m = 0.4, and Ek = 28.8: Ek = ½ mv2 The speed of the football is 12 m/s.**P. 300-303 #7-10, 11, 13, 15, 17, 18, 19-23, 27**Independent Practice