2I am not a teacher; only a fellow traveler of whom you asked the way. George Bernard Shaw ( )British dramatist, critic, writer.2
3Squaring a Number42 means 4 x 4 = 16. It is called 4 squared and 16 is called a square number.A square with sides that are 4 long has an area of 16 units242 = 16You can see the reason for calling it 4 squared and why 16 is a square number
4Perfect SquaresThe numbers 16, 36, and 49 are examples of perfect squares. A perfect square is a number that has integers as its base. Other perfect squares include 1, 4, 9, 25, 64, and 81.Number12345Square1491625Number678910Square36496481100
5Square RootsMany mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring has an inverse too, called "finding the square root."
6Square RootsRemember, the square of a number is that number times itself.The square root of a number, n, writtenis the number that gives n when multiplied by itself. For example,because 10 x 10 = 100
7Square Roots √16 means ‘the square root of 16’ and √16 = 4 A square with an area of 16 has sides that are 4 units long.Taking the square root of a number is the reverse process of squaring the number.
9Square Roots since 4 × 4 = 16. The other square root of 16 is –4, Every positive number has two square roots, one positive and one negative. One square root of 16 is 4,since 4 × 4 = 16.The other square root of 16 is –4,since (–4) × (–4) is also 16.You can write the square roots of 16 as ±4, meaning “plus or minus” 4.
10Square Roots The radical symbol returns only the positive root. This is called the principal square root of the number.To get the negative root, simply take the opposite of the principal root:
11What about negatives? Consider a negative under the radical This is asking ‘what number multiplied by itself returns a -25?’5×5 = 25(-5) ×(-5) = 25No product returns a negative valuepositivepositive
12ExampleA square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning.So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door.
13Estimating Square Roots Finding square roots of numbers that aren't perfect squares without a calculatorEstimate - first, get as close as you can by finding two perfect square roots your number is between.Divide - divide your number by one of those square roots.Average - take the average of the result of step 2 and the root.Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
143 ? 4 Estimating Square Roots Calculate the square root of 10 without a calculatorEstimate - first, get as close as you can by finding two perfect square roots your number is between.3?4Lies between 3 and 4
1510 = 3.33 3 Estimating Square Roots Calculate the square root of 10 Divide - divide your number by one of those square roots.10= 3.333Divide 10 by 3.(you can round off your answer)
163.33 + 3 = 3.1667 2 Estimating Square Roots Calculate the square root of 10Average - take the average of the result of step 2 and the root chosen in step 2.=2Average 3.33 and 3.
17Estimating Square Roots Calculate the square root of 10Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.10=Repeat step 2 with 10 and3.1667=Repeat step 3 withand2
18Estimating Square Roots Calculate the square root of 10Try the answer →Is squared equal to 10?x =If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.
19Multiplying Radicals = 2 × 3 = 6 = 6 Consider the following product Another way:= 6
20Rules of RadicalsThis process leads to a few simple rules we can use with radicals
22Simplifying RadicalsThere are three components to a simplified radical:All perfect square factors should be removed from the radicalAll fractions should be removed from the radicalAll radicals should be removed from the denominator
23Simplify RadicalsRecall we prefer to simplify fractions by removing common factors to make the denominator smaller.For radicals, we will follow a similar process, except we will concentrate on perfect square factorsPerfect Square
24Simplify Think of the largest perfect square that divides into 32 4 yes → 4×89 no16 yes → 16×2
25Simplify Think of the largest perfect square that divides into 32 4 no 9 yes → 9×5
26SimplifyEven if you can’t think of the largest perfect square, you can always simplify down through each perfect square:
27SimplifyUse the division propertyCheck:2/3 ͯ 2/3 = 4/9
28Simplify Use the division property to write as single fraction Simplify the fractionUse the division property again
29Simplify How do we remove the radical from the denominator? Recall: All radicals should be removed from the denominatorHow do we remove the radical from the denominator?
30Since this is a fraction, then let’s think about how we change the denominator of a fraction? (Without changing the value of the fraction, of course.)
31We multiply the denominator and the numerator by the same number How can we change the radical value to a rational value?
32Simply multiply the radical by itself! Remember when we square a square root, the radical goes away
33In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by
34we call this process rationalizing. Because we are changing the denominator to a rational number,we call this process rationalizing.
35SimplifyUse the division propertyRationalize the denominator
36Exponents under the Radical Variables with exponents may exist under the radical, but the same simplifying process still appliesAll even exponents are perfect squaresNotice the patternhalf the exponent
37Exponents under the Radical To simplify, we can use the perfect square propertyNo more radicalRewrite as a perfect squarehalf of 10 is 5The radical and the square offset each other
38SimplifyHalf of the exponent6 is half of 124 is half of 8
39Exponents under the Radical Odd exponents can be rewritten as an even and an odd by using the rules of exponentsRewrite odd exponent as one less (even) plus one3 = 2 + 19 = 8 + 115 =101 =
40Simplify Rewrite odd exponent as one less (even) and one half of 6 is 3the other x stays under the radical
41Adding Radicals Adding radicals is same as adding like terms: 1 + 1 = 2x + x = 2x + = 2
42Adding and Subtracting Like Radicals Add or subtract, as indicated. Assume all variables represent positive real numbers.Cannot add yet. Simplify to see if they are like radicals
43Combined Operations with Radicals You follow the same steps with these as you do with polynomials.Use the distribution property.Example: