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]
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       17

Estimate Square Roots

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

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

1. Show Solution

1. 8.7 feet