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5.01 Square Roots from Basic Algebra: One Step at a Time © 2002 P. 393-402 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
Before beginning this section on square roots, remember what it means to square a number. Squaring a number is the operation of multiplying a number times itself. A perfect square is what you get when you square a number. In other words, a perfect square is simply the square of a whole number. Before you attempt to learn and understand square roots, you must be very familiar with the perfect squares. The first few examples are given in the following list. 02 = 0 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144 132 = 169. Of course, the list goes on, but these are the important ones, the ones you will need to know and recognize in order to understand this section. Taking a square root of a number is actually the inverse operation of squaring. As subtraction is the opposite of addition, and division is the opposite of multiplication, taking a square root is the opposite of squaring. Did you ever do something and then wish you could undo it? Did you ever put on a jacket, then realize that it was too hot to be wearing a jacket? You probably un-did putting on the jacket by taking off the jacket. Did you ever do something on the computer that you didn’t mean to do? Sometimes you can un-do what you did by hitting the un-do button. Did you ever square a number like 7 (which gives you 49), and then you thought to yourself, “I wish I hadn’t done that!!” What could you do to get from 49 back to the 7 that you began with? The answer to this question is called square root, the opposite of squaring a number.
The symbol
root sign (or a radical sign). The quantity inside the square root sign is called the radicand, and the 2 is the index of the radical. In this assignment, it is important to know the perfect squares through 132 = 169. See the list above.
EXAMPLE 1.
Fill in the blanks below to find
Solution:
Therefore, the square root of 25 is 5.
EXERCISES. Fill in the blanks to find the square roots.
1.
2.
3.
4.
5.
6.
7.
8.
9.
12.
In the next exercises, you will need to remember and use the law of exponents: (xm)n = xmn . When you raise an exponent to a power, you multiply the exponents. In particular, when you square an exponent, you multiply the exponent times 2.
EXAMPLE 2. Simplify (x3)2 EXAMPLE 3. Simplify (x4)2 Solution: (x3)2 = x6 Solution: (x4)2 = x8
EXAMPLE 4.
Complete the blanks in order to find the
Solution:
EXERCISES. Complete the blanks in order to find the square roots.
14.
15.
16.
17.
18.
19.
20.
Did you notice that each time you take the square root of an exponent, you take half of the exponent, or you divide the exponent by 2? Doesn’t this seem reasonable? If you raise a power to a power, then you multiply exponents. When you take a square root (the opposite of squaring) of an exponent, then you do the opposite of “multiply exponents”, which is to “divide the exponents” by 2. Now that you have the idea, you don’t need to show all the steps.
21.
24.
In the following examples and exercises, remember that you must take the square root of the number, and divide the exponents by 2!
EXAMPLE 5.
Find the square root.
Solution:
EXAMPLE 6.
Find the square root.
Solution:
EXERCISES. Find the square roots.
27.
31.
35.
39.
43.
If the root to be taken is not a perfect power, then sometimes it can be simplified by using the product property of square roots.
Because the product property of square roots is a property of real numbers, the radicands must be positive numbers. The property does not apply if both a and b are negative numbers.
To simplify a square root by this property, it may be helpful to think of the "radical two-step":
In the first step, you must "sort" the radical, placing the "perfect squares" in the first radical and the other "leftover" factors in the second radicals. In the second step, you take the square root of the perfect square, and just bring down the "leftover" radical. The square root is simplified completely when the quantity in the radical (i.e., the radicand) is minimized. The next examples will illustrate what it means to simplify a radical expression.
EXAMPLE 7.
Simplify Step 1. Find a perfect square factor of 45, and write 45 as a product. The perfect square is 9; the “left-over” factor is 5, and the product is 9×5.
Step 2. Take the square root of 9, which is 3. Keep the square root sign on the 5.
If you wanted to
find the approximate value of nearest hundredth, you should have 6.71, which as you can see is slightly less than 7.
EXAMPLE 8. Simplify
Step 1. Find a perfect square factor of 40, and write as a product. The perfect square is 4, the leftover factor is 10, and the product is 4×10.
Step 2. Take square root of 4, which is 2. Keep the square root sign on the 10.
EXAMPLE 9.
Simplify Step 1. Find a perfect square factor of 72, and write as a product. If you said that the perfect square is 9, the leftover factor is 8, and the product is 9×8. However, this leaves a perfect square factor of 4 in the 8. You could also say the perfect square factor is 36, and the product is 36×2. If you have a choice as in this example, it is better to use the largest possible perfect square--that is, 36×2.
Step 2. Take square root of 36. Final answer:
EXERCISES. Simplify the radicals completely
47.
____________ ____________ _____________
50.
___________ ___________ ____________
53.
56.
59.
62.
65.
68.
Sometimes there are variables in the square root. As you may have noticed, when you take the square root of a variable raised to a power, you must divide the exponent by 2. If the power is even, then this is no problem. However, if the variable is raised to an odd power, then the procedure is explained in the next examples.
EXAMPLE
10. Simplify
Solution: Step 1. The perfect square factor is x2, the leftover factor is x, and you can write it as x2×x.
Step 2. Take the square root of x2, which is x.
EXAMPLE
11. Simplify
Solution: Step 1. The perfect square factors are x2 and x4. Use the largest perfect square x4, which leaves a left-over factor of x. Write x5 as the product x4×x.
Step 2. Take the square root of x4, which is x2.
EXAMPLE 12.
Simplify Solution: Step 1. The perfect square factors are 25 and the largest of the factors x2, x4, x6, x8 . Use 25x8 as the perfect square, which leaves a left-over factor of 2x. Write 50x9 as the product 25x8×2x.
Step 2. Take the square root of 25x8, which is 5x4.
EXAMPLE 13.
Simplify Solution: Step 1. First the perfect square factors are 25, x8, and since y is raised to an odd power, use one less than 5, y4. Using 25x8 y4 as the perfect square, the left-over factor is 10y. Write 250x8 y5 as 25x8 y4 × 10y.
Step 2. Take square root of 25x10y12. which is 5x5y6.
EXERCISES. Simplify the radicals completely.
71.
___________ ___________ ____________
74.
___________ ___________ ____________
77.
80.
83.
86.
89.
92.
ANSWERS 5.01 p. 393 - 402: 1.
7; 2. 12; 3. 3; 4. 4; 5. 8; 6. 2; 7.
0; 8. 1; 9. 9; 10. 11; 11. 10; 12.
6; 13. 13; 14. x2; 15. x3;
16. x4; 17. x5; 18. x6;
19. x7; 20. x10; 21. x12;
22. x14; 23. x24; 24. x25;
25. x18; 26. x50; 27. 7x3;
28. 6x2; 29. 5x6; 30. 2x5;
31. 4x3; 32. 7x6; 33. 6x18;
34. 5x50; 35. 3x9; 36. 13x11;
37. 6x7; 38. 8x4; 39. 5x10;
40. 4x40; 41. 10x15; 42. 11x;
43. 9x8; 44. 12x2; 45. 8x50;
46. 6x72; 47. Return to main page Math in Living C O L O R !! Return to Basic Algebra page Return to Intermediate Algebra page
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