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