Math- Lesson - Radicals
= What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent number The decimal, is just an approimation.
Radicals Inde number Radical symbol Radicand = = = = = = The square root of means: what number squared equals? The rd root of means: what number cubed equals? The th root of means: what number used as a factor times equals?
Adding and subtracting radicals Can these two terms be combined using addition? + + + + + + + + Write as repeated addition Write as repeated addition + When multiplication is written as repeated addition, like terms look eactly alike. + + + + + + + + + +
Define like powers + Define like radicals + Same base, same eponent. Same radicand, same inde number. Which of the following are like radicals that can be added? + + + + + +
+ + = Are they equivalent? 1.71... 1.1... +.1....0... + a + b a + b This is NOT a property of radicals. NEVER DO THIS!!!! + 9 1 + 9 + 1
Simplify the following: + 8 + 7 + 9 + + + 7 + not like terms in their present form 7
Will this work? 1.71... 1.1....9.9... Product of Radicals Property a b a b = 10 9 9 Are these equivalent? a b = ab = Although I only gave two eamples, it actually DOES WORK for whole number radicand. 8
a b = ab Simplify the following: 8 1 1 8 1 8 8 1 = 1 0 7 7 + 1 + 9 0 9
Simplify radicals: use the Product of Radicals to break apart the radical into a perfect square times a number. a b = ab 18 Simplify 9 + 1 1 y 8 7y 7y 1y
Stop Here
Can we add unlike radicals?. a b = ab Simplify 7 + ( 7 + ) 7 + ( ) 7 + 11 + 8 ( 1 ) + ( ) + ( ) ( ) 1 + 8
Simplify radicals: use the Product of Radicals to break apart the radical into a powers of eponent m times a number. Simplify y m m a b = m ab 1 8 y y y y y y y y y
Another way to Simplify Radicals 7 9 Factor, factor, factor!!! What is the factor that is used times under the radical? Bring that out factor (that is used times). Using Properties of Eponents to reduce the writing: 1
Factor the numerator! 1 Inverse Property of Multiplication 8 1 1 1 Inverse Property of Multiplication y 8y 7 8y 8y Inverse Property of Multiplication 7 7 1
Simplify 9 0y y 9 7 8 9 1
Rationalizing the denominator: using mathematical properties to change an irrational number (or imaginary) in the denominator into a rational number. We take advantage of the idea: = = = = 9 = 1 Identity Property of Multiplication = multiplying by 1 doesn t change the number.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 7 9 1
In all of the previous eamples we just multiplied by one in the form of the denominator radical over the denominator radical. 7 8 7 8 8 8 It is always easier to simplify (by factoring) BEFORE you multiply 7 8 7 8 1 7 8 7 7 1 1 1 1
What about variables? 1
What about higher inde numbers? 1 How many more s are needed in the denominator radicand? Remember: the cubed root of -cubed equals. We need two more s under the denominator radical. = Using the multiply powers property we don t have to write out all the individual s. 1
Rationalize the denominator y y
What about higher inde numbers? 1 1 1 How many more s and s are needed in the denominator radicand? We need one more and two more s under the denominator radical. 1
What about higher inde numbers? How many more s and s are needed in the denominator radicand? We need one more s and one more under the denominator radical. 1