RADICAL EXPRESSIONS
ARE THE SAME AS SQUARE ROOTS RADICAL EXPRESSIONS ARE THE SAME AS SQUARE ROOTS
Radical Expressions Square Roots & Perfect Squares Simplifying Radicals Multiplying Radicals Dividing Radicals Adding Radicals
What is the square root? 64
The square root of a number is a # when multiplied by itself equals the number 64 8 8
What is the square root? X2
A Square Root is a term when squared is the Perfect Square X2 X X
What is the square root? 64x2
Answer 64x2 8x 8x
What is the square root? 4x2
Answer 4x2 2x 2x
What is the square root? (x+3)2
Answer (x+3) (x+3)2 (x+3)
Radical Sign • This is the symbol for square root • If the number 3 is on top of the v, it is called the 3rd root.
Radical Sign • This is the 4th root. • This is the 6th root • If n is on top of the v, it is called the nth root.
Cube Root If the sides of a cube are 3 inches Then the volume of the cube is 3 times 3 times 3 or 33 which is 27 cubic inches.
NUMBER Perfect Squares Numbers that are perfect squares are: 12=1, 22= 4, 32= 9, 42= 16, 52= 25, 62= 36, 72= 49, 82= 64, 92= 81, 102= 100, …
Recognizing Perfect Squares (NAME THE SQUARE ROOTS) Variables that are perfect squares are: x2, a4, y22, x100… (Any even powered variable is a perfect square) x25 EVEN POWERED EXPONENTS ARE SQUARES x25 x50
NAME THE SQUARE ROOT 9x50
3x25 EVEN POWERED EXPONENTS ARE SQUARES 3x25 9x50
Recognizing Perfect Squares (NAME THE SQUARE ROOTS) x2 4 25x2 4x6
Recognizing Perfect Squares (NAME THE SQUARE ROOTS) x 2 x x2 2 4 5x 2x3 5x 25x2 2x3 4x6
Simplifying Square Roots
Simplifying Square Roots We can simplify if 8 contains a Perfect Square
Simplifying Square Roots Look to factor perfect squares (4, 9, 16, 25, 36…) Put perfect squares in first radical and the other factor in 2nd. Take square root of first radical
Simplifying Square Roots
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) (Put perfect squares in first radical and other factor in 2nd)
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Square Roots
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd)
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd)
Simplifying Radicals 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd)
Simplifying Radicals 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Fraction Radicals
Simplifying Fraction Radicals 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM. (& Put perfect squares in first radical and other factor in 2nd)
Simplifying Fraction Radicals 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM. (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Fraction Radicals 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM. (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical 3. REDUCE
Finding Perfect Squares The most common perfect squares are 4 & 9. USE DIVISIBILITY RULES: A number is divisible by 4 if the last 2 digits are divisible by 4. ( 23,732 is divisible by 4 since 32 is.) A number is divisible by 9 if the sum of it’s digits are. (4653 is divisible by 9 since the sum of it’s digits are.)
Finding Perfect Squares MORE EASY TO FIND PERFECT SQUARES: A number with an even number of Zeros. ( 31,300 is divisible by 10, 70,000 is divisible by 100) A number ending in 25, 50 or 75 is divisible by 25. (425 & 350 & 775 are all divisible by 25)
Perfect Squares Guide Divisibility Rules for Perfect Squares 4: If Last 2 Digits are Divisible by 4 9: If Sum of Digits are Divisible by 9 16: Use 4 Rule 25: If # ends in 25, 50, 75 or 00 36: Use 4 or 9 Rule 49: Look for multiple of 50 and subtract multiple 64: Use 4 Rule 81: Use 9 Rule 100: If # ends in 00 121: No Rule, Divide # by 121 144: Use 4 or 9 Rule
Multiply Radicals
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…)
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…)
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…) • Take square root of first radical
Multiply Radicals
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…)
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…) • Take square root of first radical
Multiply Radicals
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…)
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…) • Take square root of first radical
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…)
Multiply Radicals Multiply to one radical Simplify • Factor any perfect squares (4, 9, 16, x2, y6…) • Take square root of first radical
Another Way to Multiply Radicals
Multiply Radicals By Factoring Perfect Squares First
Multiply Radicals By Factoring Perfect Squares First Factor any perfect squares
Multiply Radicals By Factoring Perfect Squares First Factor any perfect squares
Multiply Radicals By Factoring Perfect Squares First Factor any perfect squares Take square roots of perfect squares
Multiply Radicals By Factoring Perfect Squares First Factor any perfect squares Take square roots of perfect squares Multiply #,s &
Multiply Radicals with Distributive Prop.
Multiply Radicals with Distributive Prop. Mult. With Distr. Property
Multiply Radicals with Distributive Prop. Mult. With Distr. Property
Multiply Radicals with Distributive Prop. Mult. With Distr. Property Factor any perfect squares
Multiply Radicals with Distributive Prop. Mult. With Distr. Property Factor any perfect squares Take square roots of perfect squares Simplify if possible
Multiply Radicals with Distributive Prop. Multiply Terms (Use Multiplying Boxes & CLT) Simplify if possible x 3x 3x2
Dividing Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical. 3. Reduce
Dividing Square Roots
Dividing Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator (& Put perfect squares in first radical and other factor in 2nd)
Dividing Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator (& Put perfect squares in first radical and other factor in 2nd)
Dividing Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical.
Dividing Square Roots 1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical. 3. Reduce
Divide Radicals
Divide Radicals Divide
Divide Radicals Divide
Divide Radicals Divide Simplify
Divide Radicals Divide Simplify
Divide Radicals Rationalizing the Denominator
Divide Radicals Rationalizing the Denominator Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator
Divide Radicals Rationalizing the Denominator Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator
Divide Radicals Rationalizing the Denominator Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator Take the square root of the Perfect Square Simplify
Divide Radicals Rationalizing the Denominator Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator Take the square root of the Perfect Square Simplify
Divide Radicals Rationalizing the Denominator(2 term) Rationalizing the denominator means making the denominator rational or REMOVING IRRATIONAL TERMS
Divide Radicals Rationalizing the Denominator(2 term) Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator The conjugate of a binomial is SWITCHING THE SIGN BETWEEN THEM
Divide Radicals Rationalizing the Denominator(2 term) Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator 8 8 64
Divide Radicals Rationalizing the Denominator(2 term) Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator Take the square root of the Perfect Square
Divide Radicals Rationalizing the Denominator(2 term) Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator Take the square root of the Perfect Square Simplify
Square Roots of Perfect Squares 1. Factor any perfect squares 2. Take square root of perfect square.
Values that make it Real 1. Set term in sq. root ≥ 0 2. Solve
Values that make it Real
Values that make it Real 1. Set term in sq. root ≥ 0 2. Solve
Values that make it Real Just note that if x is any real number (all numbers on the number line) x2 can never be negative, so (x2+16) will always be positive. The answer is any real number
Negative Square Roots Can you take the square root of a negative number?
Can you take the square root of a negative number? NO Because a negative number squared is positive
Simplifying Square Roots 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical
Simplifying Negative Square Roots 1. Factor any perfect squares (4, 9, 16, 25, 36…) (& Put perfect squares in first radical and other factor in 2nd) 2. Take square root of first radical 3. The square root of -1 is called IMAGINARY (We will review this later)
Adding Radicals
Adding Radicals Same as Adding LIKE TERMS
As Simple As
As Simple As + =
Similar Radical's are Like Terms + =
Can't Combine Unlike Terms + = + a + b = a + b
Only Combine Like Terms + = + + a + a + b = 2a + b
Always Combine Like Terms + = a + a = 2a
Always Combine Like Terms + = a + a = 2a
As Simple As
As Simple As
one radical x equals two radical x's As Simple As one radical x plus one radical x equals two radical x's
Same as Adding LIKE TERMS
Same as Adding LIKE TERMS
Same as Adding LIKE TERMS
Same as Adding LIKE TERMS
Same as Adding LIKE TERMS
Same as Adding LIKE TERMS
Can you add unlike terms?
NO You can't combine unlike terms
Try This One
Answer
Write Down Your Answers
Answers
Write Down Your Answers
Answers
Adding Radicals
Adding Radicals Simplify to Like
Adding Radicals Simplify to Like
Adding Radicals Simplify to Like Add the Like
Adding Radicals
Adding Radicals Simplify to Like • GCF
Adding Radicals Simplify to Like • GCF
Adding Radicals Simplify to Like • GCF •
Adding Radicals Simplify to Like • GCF • •Simplify
Adding Radicals Simplify to Like • GCF • •Simplify Add the Like Terms
Adding Radicals
Adding Radicals Simplify to Like • Rationalize Denominator
Adding Radicals Simplify to Like • Rationalize Denominator
Adding Radicals Simplify to Like • Rationalize Denominator Add the Like Terms