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Square Roots “ ^ “ plus any number equals exponent “ { “ equals square root symbol Perfect Squares To find a perfect square of a number you take a whole number that is between 1 and 10 and multiply it by itself. Square of n = n^2 Square of 5= 5x5= 5^2= 25 The perfect squares are the squares of the whole numbers. Perfect Squares: 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 Finding Square Roots of Numbers That Aren't Perfect Squares Without a Calculator To find the square root of a number without using a calculator you must do the following steps: 1. Estimate - first, get as close as you can by finding two perfect square roots your number is between. 2. Divide - divide your number by one of those square roots. 3. Average - take the average of the result of step 2 and the root. 4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you. Example Calculate the square root of 10 ({10) to 2 decimal places. 1. Find the two perfect square numbers it lies between. Solution: 32 = 9 and 42 = 16, so {10 lies between 3 and 4. 2. Divide 10 by 3. 10/3 = 3.33 3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667 Repeat step 2: 10/3.1667 = 3.1579 Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623 Check Your Answer - Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001 Practice Problems for Solving Square Roots Without Using a Calculator Problem#1: 60 Problem#2: 7 Problem#3: 75 Problem#4: 33 Problem#5: 66 Answer to Problem#1: 7.745 Answer to Problem#2: 2.645 Answer to Problem#3: 8.660 Answer to Problem#4: 5.744 Answer to Problem#5: 8.124 Properties of square roots Square roots have the following properties {a+b = {a + {b {ab = {a * {b {a/b = {a/{b [{a]^2 = a Addition and Subtraction of Square Roots Adding and subtracting square roots is just like combining like terms when you need to do that with algebraic expressions. If the indices (a square root's index is 2, a cube root's index is 3, a 4th root's index is 4, etc.) or the radicands (the expression under the root sign or enclosed by brackets after the square root) are the same. When adding and subtracting you keep the bases the same and add or subtract just the co-efficient. Examples 4{2 + 7{2 = 11{2 12{3 – 8{3 = 4{3 2{2 – 5{2 + 67{2 = 64{2 Practice Problems Adding and Subtracting Square Roots Problem#1: 95{95 + 95{95= Problem#2: 5{6 + 36{6= Problem#3: 65{7 – 23{7= Problem#4: 33{66 + 1{66 – 235{66= Problem#5: 854{32 – 855{32 + {32= Quick Solutions Answer to Problem#1: 190{95 Answer to Problem#2: 41{6 Answer to Problem#3: 42{7 Answer to Problem#4: -201{66 Answer to Problem#5: 0 Multiplication and Division of Square Roots To multiply a square root equation you multiply all of the co-efficients together and then all of the radicals together. To divide a square root equation you have to take the two terms and divide the co-efficients together and then all of the radicals together. Examples 2. 25{9/5{3= 25/5=5= {9/{3={3= 5{3 2. 2{6*62{2= 2*62=124= {6*{2={12 124{12 Practice Problems for Multiplication and Division Problem#1: 6{4*7{8= Problem#2: 12{4/3{2= Problem#3: 81{24/9{3= Problem#4: 126{45*2{3= Problem#5: 7{4^2= Quick Solutions Answer to Problem#1: 42{32 Answer to Problem#2: 36{8 Answer to Problem#3: 9{8 Answer to Problem#4: 63{15 Answer to Problem#5: 49{16 Mini Quiz 1. Is 75 a perfect square? 2. What is the root of 64? 3. {30= 4. {52= 5. 3{6 + 4{6= 6. 16{2 – 11{2= 7. 22{7*8{3= 8. 33{16/3{4= 9. 15{7^2= 10. 8{3^2 + 6{3^2= Quick Solutions to Mini Quiz 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. No 8 5.477 7.211 7{6 5{2 176{21 11{4 30{7 100{9 Websites For Further Information http://library.thinkquest.org/20991/textonly/alg/ro ots.html?tqskip1=1&tqtime=0410 http://math.usask.ca/readin/roots.html http://www.gomath.com/htdocs/lesson/radical_lesso n3.htm www.purple.com