Simplify Expressions with Square Roots

In the following exercises, simplify.

1. 1. [latex]\sqrt{36}[/latex]
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      6
   2. [latex]\sqrt{4}[/latex]
   3. [latex]\sqrt{64}[/latex]
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      8
   4. [latex]\sqrt{144}[/latex]
   5. [latex]-\sqrt{4}[/latex]
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      −2
   6. [latex]-\sqrt{100}[/latex]
   7. [latex]-\sqrt{1}[/latex]
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      −1
   8. [latex]-\sqrt{121}[/latex]
   9. [latex]\sqrt{-121}[/latex]
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      not a real number
   10. [latex]\sqrt{-36}[/latex]
   11. [latex]\sqrt{-9}[/latex]
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       not a real number
   12. [latex]\sqrt{-49}[/latex]
   13. [latex]\sqrt{9+16}[/latex]
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       5
   14. [latex]\sqrt{25+144}[/latex]
   15. [latex]\sqrt{9}+\sqrt{16}[/latex]
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       7
   16. [latex]\sqrt{25}+\sqrt{144}[/latex]
   17. [latex]\sqrt{64}[/latex]
   18. [latex]\sqrt{144}[/latex]
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       12
   19. [latex]-\sqrt{25}[/latex]
   20. [latex]-\sqrt{81}[/latex]
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       −9
   21. [latex]\sqrt{-9}[/latex]
   22. [latex]\sqrt{-36}[/latex]
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       not a real number
   23. [latex]\sqrt{64}+\sqrt{225}[/latex]
   24. [latex]\sqrt{64+225}[/latex]
       Show Solution

       17

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

1. 1. [latex]\sqrt{70}[/latex]
      Show Solution

      [latex]8<\sqrt{70}<9[/latex]
   2. [latex]\sqrt{55}[/latex]
   3. [latex]\sqrt{200}[/latex]
      Show Solution

      [latex]14<\sqrt{200}<15[/latex]
   4. [latex]\sqrt{172}[/latex]
   5. [latex]\sqrt{28}[/latex]
   6. [latex]\sqrt{155}[/latex]
      Show Solution

      [latex]12<\sqrt{155}<13[/latex]

Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

1. 1. [latex]\sqrt{19}[/latex]
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      4.36
   2. [latex]\sqrt{21}[/latex]
   3. [latex]\sqrt{53}[/latex]
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      7.28
   4. [latex]\sqrt{47}[/latex]
   5. [latex]\sqrt{15}[/latex]
   6. [latex]\sqrt{57}[/latex]
      Show Solution

      7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

1. 1. [latex]\sqrt{{y}^{2}}[/latex]
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      y
   2. [latex]\sqrt{{b}^{2}}[/latex]
   3. [latex]\sqrt{49{x}^{2}}[/latex]
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      7x
   4. [latex]\sqrt{100{y}^{2}}[/latex]
   5. [latex]-\sqrt{64{a}^{2}}[/latex]
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      −8a
   6. [latex]-\sqrt{25{x}^{2}}[/latex]
   7. [latex]\sqrt{144{x}^{2}{y}^{2}}[/latex]
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      12xy
   8. [latex]\sqrt{196{a}^{2}{b}^{2}}[/latex]
   9. [latex]\sqrt{{q}^{2}}[/latex]
   10. [latex]\sqrt{64{b}^{2}}[/latex]
       Show Solution

       8b
   11. [latex]-\sqrt{121{a}^{2}}[/latex]
   12. [latex]\sqrt{225{m}^{2}{n}^{2}}[/latex]
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       15mn
   13. [latex]-\sqrt{100{q}^{2}}[/latex]
   14. [latex]\sqrt{49{y}^{2}}[/latex]
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       7y
   15. [latex]\sqrt{4{a}^{2}{b}^{2}}[/latex]
   16. [latex]\sqrt{121{c}^{2}{d}^{2}}[/latex]
       Show Solution

       11cd

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of [latex]75[/latex] square feet. How long can a side of his garden be?

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. How long can a side of his patio be?

Gravity An airplane dropped a flare from a height of 1,024 feet above a lake. How many seconds did it take for the flare to reach the water?

Gravity A hiker dropped a granola bar from a lookout spot 576 feet above a valley. How long did it take the granola bar to reach the valley floor?

Gravity A hang glider dropped his cell phone from a height of 350 feet. How many seconds did it take for the cell phone to reach the ground?

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4,000 feet above the Colorado River. How many seconds did it take for the hammer to reach the river?

Accident investigation The skid marks from a car involved in an accident measured 54 feet. What was the speed of the car before the brakes were applied?

Accident investigation The skid marks from a car involved in an accident measured 216 feet. What was the speed of the car before the brakes were applied?

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. What was the speed of the vehicle before the brakes were applied?

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. What was the speed of the vehicle before the brakes were applied?

2. Everyday Math

   1. Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?
   2. Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2,025 tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?
      Show Solution

      45 inches

   Writing exercises

   • Why is there no real number equal to [latex]\sqrt{-64}[/latex]?
   • What is the difference between 9² and [latex]\sqrt{9}[/latex]?
     Show Answer

     Answers will vary. 92 reads: “nine squared” and means nine times itself. The expression [latex]\sqrt{9}[/latex] reads: “the square root of nine” which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.