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Radicals I am not a teacher; only a fellow traveler of whom you asked the way. George Bernard Shaw (1856-1950) British dramatist, critic, writer. Squaring a Number 42 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 units2 42 = 16 You can see the reason for calling it 4 squared and why 16 is a square number Perfect Squares The 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. Number Square Number Square 1 2 3 4 5 1 4 9 16 25 6 7 8 9 10 36 49 64 81 100 Square Roots Many 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." Square Roots Remember, the square of a number is that number times itself. The square root of a number, n, written n is the number that gives n when multiplied by itself. For example, 100 10 because 10 x 10 = 100 Square 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. Square root 1 = 1 Perfect square 81 Square root 81 = 9 4 4 = 2 100 100 = 10 9 9 = 3 121 121 = 11 16 16 = 4 144 144 = 12 25 25 = 5 169 169 = 13 36 36 = 6 196 196 = 14 49 49 = 7 225 225 = 15 64 64 = 8 Perfect square 1 Square Roots 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. Square 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: 25 5 25 5 What about negatives? Consider a negative under the radical 25 This is asking ‘what number multiplied by itself returns a -25?’ 5×5 = 25 positive (-5) ×(-5) = 25 positive No product returns a negative value Example A 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. 16 4 So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door. Estimating Square Roots Finding square roots of numbers that aren't perfect squares without a calculator 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. Estimating Square Roots Calculate the square root of 10 without a calculator 10 1. Estimate - first, get as close as you can by finding two perfect square roots your number is between. 9 10 16 3 ? 4 10 Lies between 3 and 4 Estimating Square Roots Calculate the square root of 10 10 2. Divide - divide your number by one of those square roots. 3 10 4 10 = 3.33 3 Divide 10 by 3. (you can round off your answer) Estimating Square Roots Calculate the square root of 10 10 3. Average - take the average of the result of step 2 and the root chosen in step 2. 3.33 + 3 = 3.1667 2 Average 3.33 and 3. Estimating Square Roots Calculate the square root of 10 10 4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you. Repeat step 2 with 10 and 3.1667 10 = 3.1579 3.1667 3.1667 + 3.1579 = 3.1623 Repeat step 3 with 3.1667 and 3.1579 2 Estimating Square Roots Calculate the square root of 10 10 Try the answer → Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001 If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3. 10 3.1623 Multiplying Radicals Consider the following product 4 9 = 2 × 3 =6 Another way: 4 9 4 9 36 =6 Rules of Radicals This process leads to a few simple rules we can use with radicals m 2 1. m 2. m n m n 3. m n 2 m n m Multiply 5 2 5 2 10 3 3 3 3 9 3 Simplifying Radicals There are three components to a simplified radical: 1. All perfect square factors should be removed from the radical 2. All fractions should be removed from the radical 3. All radicals should be removed from the denominator Simplify Radicals Recall 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 factors Perfect Square 8 4 2 4 2 2 2 Simplify Think of the largest perfect square that divides into 32 4 yes → 4×8 9 no 16 yes → 16×2 32 16 4 2 2 Simplify 45 Think of the largest perfect square that divides into 32 4 no 9 yes → 9×5 9 3 5 5 Simplify Even if you can’t think of the largest perfect square, you can always simplify down through each perfect square: 4 6 96 4 24 4 6 2 2 6 Simplify Use the division property Check: 2/3 ͯ 2/3 = 4/9 4 4 9 9 2 3 Simplify Use the division property to write as single fraction Simplify the fraction Use the division property again 8 8 50 50 4 25 4 2 25 5 Simplify Recall: All radicals should be removed from the denominator 1 3 How do we remove the radical from the denominator? Since 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.) 1 3 We multiply the denominator and the numerator by the same number 1 1 3 3 4 4 3 12 How can we change the radical value to a rational value? Simply multiply the radical by itself! 2 3 3 3 3 Remember when we square a square root, the radical goes away m 2 m In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by 3 1 1 3 3 3 3 3 3 Because we are changing the denominator to a rational number, we call this process rationalizing. 3 1 3 3 Simplify Use the division property Rationalize the denominator 3 3 5 5 3 5 15 5 5 5 Exponents under the Radical Variables with exponents may exist under the radical, but the same simplifying process still applies All even exponents are perfect squares Notice the pattern half the exponent x x 2 2 x x x x 4 x 6 x 8 2 2 3 2 4 2 4x 2 Exponents under the Radical To simplify, we can use the perfect square property 2 m m Rewrite as a perfect square 10 half of 10 is 5 x x 5 2 No more radical x 5 The radical and the square offset each other Simplify 4x 2 12 x ( x 1) 8 2x Half of the exponent x 6 is half of 12 6 ( x 1) 4 4 is half of 8 Exponents under the Radical Odd exponents can be rewritten as an even and an odd by using the rules of exponents Rewrite odd exponent as one 3 2 3=2+1 x x x less (even) plus one 8 9 x x x 9=8+1 x 15 x x 15 = 14 + 1 x 101 x 101 = 100 + 1 14 100 x Simplify Rewrite odd exponent as one less (even) and one half of 6 is 3 x 7 x x x 6 3 the other x stays under the radical x Adding Radicals Adding radicals is same as adding like terms: 1+1=2 x + x = 2x +=2 3 3 2 3 Adding and Subtracting Like Radicals Add or subtract, as indicated. Assume all variables represent positive real numbers. 7 2 8 18 4 72 Cannot add yet. Simplify to see if they are like radicals 7 2 8 18 4 72 7 2 8 9 2 4 36 2 7 2 83 2 46 2 7 2 24 2 24 2 7 2 Combined Operations with Radicals You follow the same steps with these as you do with polynomials. Use the distribution property. Example: 2 ( 10 2 ) 20 4 2 52