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Compare the following pairs of numbers using <, >, or =. Show all work.

Mathematics

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Answers

8 parts
a

<<

b

>>

c

>>

d

<<

e

==

f

<<

g

<<

h

<<

Step-by-step solution

24 steps

Here are the comparisons:

Part a· 3 steps

  1. Evaluate 222^2.

    22=2×2=42^2 = 2 \times 2 = 4

  2. Evaluate 525^2.

    52=5×5=255^2 = 5 \times 5 = 25

  3. Compare the values.

    4<254 < 25

    Therefore, 22<522^2 \boxed{<} 5^2.

Part b

>>

  1. Evaluate 252^5.

    25=2×2×2×2×2=322^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

  2. Evaluate 525^2.

    52=5×5=255^2 = 5 \times 5 = 25

  3. Compare the values.

    32>2532 > 25

    Therefore, 25>522^5 \boxed{>} 5^2.

Part c

>>

  1. Evaluate 636^3.

    63=6×6×6=2166^3 = 6 \times 6 \times 6 = 216

  2. Evaluate 232^3.

    23=2×2×2=82^3 = 2 \times 2 \times 2 = 8

  3. Compare the values.

    216>8216 > 8

    Therefore, 63>236^3 \boxed{>} 2^3.

Part d

<<

  1. Evaluate 525^2.

    52=5×5=255^2 = 5 \times 5 = 25

  2. Evaluate 333^3.

    33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

  3. Compare the values.

    25<2725 < 27

    Therefore, 52<335^2 \boxed{<} 3^3.

Part e

==

  1. Evaluate 202^0.

    20=12^0 = 1

  2. Evaluate 404^0.

    40=14^0 = 1

  3. Compare the values.

    1=11 = 1

    Therefore, 20=402^0 \boxed{=} 4^0.

Part f

<<

  1. Evaluate 64\sqrt{64}.

    64=8\sqrt{64} = 8

  2. Evaluate 323^2.

    32=3×3=93^2 = 3 \times 3 = 9

  3. Compare the values.

    8<98 < 9

    Therefore, 64<32\sqrt{64} \boxed{<} 3^2.

Part g

<<

  1. Evaluate 643\sqrt[3]{64}.

    643=4\sqrt[3]{64} = 4

  2. Evaluate 64\sqrt{64}.

    64=8\sqrt{64} = 8

  3. Compare the values.

    4<84 < 8

    Therefore, 643<64\sqrt[3]{64} \boxed{<} \sqrt{64}.

Part h

<<

  1. Consider the first number, 3.

  2. Estimate 284\sqrt{284}.

    We know that 162=25616^2 = 256 and 172=28917^2 = 289.
    Since 256<284<289256 < 284 < 289, it means 16<284<1716 < \sqrt{284} < 17.

  3. Compare 3 with a number between 16 and 17.

    3<2843 < \sqrt{284}

    Therefore, 3<2843 \boxed{<} \sqrt{284}.

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