Simplifying Radical Expressions
Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd, then n ab n a n b
1. 144x 8 y 5 12 2 (x 4 ) 2 (y 2 ) 2 y 12 2 (x 4 ) 2 (y 2 ) 2 y 12x 4 y 2 y Factor into squares Product Property of Radicals
Quotient Property of Radicals For real numbers a and b, b 0, And any integer n, n>1, a n n a b n b, if all roots are defined. Ex: 81 256 81 256 9 16
In general, a radical expression is simplified when: The radicand contains no fractions. No radicals appear in the denominator.(rationalization) The radicand contains no factors that are nth powers of an integer or polynomial.
Simplify each expression. 1) x 6 y 3 x 3 y y y y Rationalize the denominator x 6 x3 x 3 y 3 y 2 y x3 y y x 3 y y y x 3 y 2 y Answer
To simplify a radical by adding or subtracting you must have like terms. Like terms are when the powers AND radicand are the same. Ex: 3 5 and 6 3 5, 2x 6z and 5x 6z
Here is an example that we will do together. 3 20 150 5 45 3 2 2 5 5 2 6 5 3 2 5 Rewrite using factors 3 2 5 5 6 5 3 5 6 5 5 6 15 5 9 5 5 6 Combine like terms
You can add or subtract radicals like monomials. You can also simplify radicals by using the FOIL method of multiplying binomials. Let us try one. Ex: (3 6 2 3)(4 3)
(3 6 2 3)(4 3) F 3 6 O I L 4 3 6 3 2 3 4 2 3 3 12 6 3 3 2 2 8 3 2 3 12 6 9 2 8 3 6 Since there are no like terms, you can not combine.
Rational Exponents In other words, exponents that are fractions.
Laws of Exponents Negative Exponents: a n 1 a n 1 a n a n Multiplying Powers: a m x a n a mn
Laws of Exponents Dividing Powers: a a m n a mn Power of a Power: m a n a mn Power of a Product: ab m a m b m
Laws of Exponents Power of a Quotient: a b n a b n n b n a n a b n Power of Zero: 0 a 1
1 Definition of b n For any real number b and any integer n > 1, b 1 n n b except when b < 0 and n is even
Examples: 1 1. 362 36 6 1 2. 64 3 3 64 4
Definition of Rational Exponents For any nonzero number b and any integers m and n with n > 1, m n m b n n b m b except when b < 0 and n is even
NOTE: There are 3 different ways to write a rational exponent 4 3 4 27 3 3 27 4 27
Simplifying Expressions No negative exponents No fractional exponents in the denominator No complex fractions (fraction within a fraction) The index of any remaining radical is the least possible number
Examples: Simplify each expression 1 3 1 1 x 2 x 4 x 5 x 2 3 4 1 5 x 10 15 4 20 20 20 x 29 20 Remember we add exponents 20 9 x20 x x x 20 20 9
Examples: Simplify each expression 5 1 2 2 5 1 5 2 1 2 5 1 2 1 2 5 1 2 5 2 12 1 2 51 1 2 1 5 1 10 5 1 2 1 2
Solve radical equations having square root radicals. To solve radical equations having square root radicals, we need a new property, called the squaring property of equality. If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation. Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation.
Using the Squaring Property of Equality Solve. Solution: 9x 4 2 2 9 x 4 9x 16 9x 916 9 x 7 7 7 x It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.
Using the Squaring Property with a Radical on Each Side Solve. 3x9 2 x Solution: 3x9 2 x 2 2 3x9 4x 3x93x 4x3x x 9 9
Solving a Radical Equation. Use the following steps when solving an equation with radicals. Step 1 Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Step 2 Step 3 Step 4 Step 5 Step 6 Square both sides. Combine like terms. Repeat Steps 1-3 if there is still a term with a radical. Solve the equation. Find all proposed solutions. Check all proposed solutions in the original equation.
Using the Squaring Property with a Quadratic Expression Solve 2 x x x 4 16. Solution: 4 16 2 2 2 x x x x x x 4x 16 x 2 2 2 2 04x 4x 16 4x 4x 16 4 4 x 4 Since x must be a positive number the solution set is Ø.
Using the Squaring Property when One Side Has Two Terms 2x1 10x 9. Solution: 2 2x1 10x 9 2 2 4x 4x 1 10x 9 10x 9 10x 9 2 4x 14x8 0 2x1 2x8 0 2x 10 or 2x 8 0 1 x 2 x 4 Since x must be positive the solution set is {4}. Slide 8.6-27
Exponential Equations with Like Bases In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like 3 2x1 5 4 or 3 x1 9 x2 If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple.
Exponential Growth Function If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation Where C = initial amount y C( 1 r) t r = growth rate (percent written as a decimal) t = time where t 0 (1+r) = growth factor where 1 + r > 1
Example Compound Interest You deposit $1500 in an account that pays 2.3% interest compounded yearly, 1) What was the initial principal (P) invested? 2) What is the growth rate (r)? The growth factor? 3) Using the equation A = P(1+r) t, how much money would you have after 2 years if you didn t deposit any more money? 1) The initial principal (P) is $1500. 2) The growth rate (r) is 0.023. The growth factor is 1.023. 3) A P ( 1 r) A 1500( 1 0. 023) A $ 1569. 79 t 2
If a quantity decreases by the same proportion r in each unit of time, then the quantity displays exponential decay and can be modeled by the equation Where Exponential Decay Function C = initial amount y C( 1 r) t r = growth rate (percent written as a decimal) t = time where t 0 (1 - r) = decay factor where 1 - r < 1
Example Depreciation Interest You buy a new car for $22,500. The car depreciates at the rate of 7% per year, 1) What was the initial amount invested? 2) What is the decay rate? The decay factor? 3) What will the car be worth after the first year? The second year? 1) The initial investment was $22,500. 2) The decay rate is 0.07. The decay factor is 0.93. 3) y C( 1 r) y y 22, 500( 1 0. 07) $ 20, 925 t 1 y C( 1 r) y y 22, 500( 1 0. 07) $ 19460. 25 t 2
2) Your business had a profit of $25,000 in 1998. If the profit increased by 12% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve.
C = $25,000 T = 12 R = 0.12 Growth factor = 1.12 y C( 1 r) y y y t $ 25, 000( 1 0. 12) $ 25, 000( 112. ) $ 97, 399. 40 12 12
3) Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, when would there be 1.56mg left? Identify C, t, r, and the decay factor. Write down the equation you would use and solve.
1.56y C( 1 r) 1.56y 1.56y C = 25 mg T =? R = 0.5 Decay factor = 0.5 y = 1.56 mg 25mg( 1 0. 5) 25mg( 0. 5) t.0624 y = 10.5. 56mg t 4t t4 Use the calculator table
Exponential Equations with Like Bases Example #1 - One exponential expression. 3 2 x1 5 4 3 2 x1 9 3 2 x1 3 2 2x 1 2 2x 1 1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation. x 1 2
Exponential Equations with Like Bases Example #2 - Two exponential expressions. 3 x1 9 x2 3 x1 3 2 x2 3 x1 3 2 x 4 x 1 2x 4 x 5 1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.
Logarithmic Equations Example 1 - Variable inside the log function. log 4 2x 1 3 5 log 4 2x 1 2 4 2 2x 1 16 2x 1 2x 17 x 8.5 1. Isolate the log expression. 2. Rewrite the log equation as an exponential equation and solve for x.
Logarithmic Equations Example 3 - Variable inside the base of the log. log x 3 2 x 2 3 1. Rewrite the log equation as an exponential equation. x 2 1 2 3 1 2 2. Solve the exponential equation. x 1 3 x 3 3