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Simplify 8¹/3.

Mathematics

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Answers

2 parts
i

18\frac{1}{8}

ii

181\frac{1}{81}

Step-by-step solution

10 steps

Part i· 5 steps

  1. Simplify 81/38^{1/3}.

    81/3=83=28^{1/3} = \sqrt[3]{8} = 2.

  2. Rewrite the expression using the simplified term and express all numbers as powers of 2.

    81/3×8−3×32=2×(23)−3×258^{1/3} \times 8^{-3} \times 32 = 2 \times (2^3)^{-3} \times 2^5

  3. Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}.

    2×23×(−3)×25=21×2−9×252 \times 2^{3 \times (-3)} \times 2^5 = 2^1 \times 2^{-9} \times 2^5

  4. Apply the exponent rule am×an=am+na^m \times a^n = a^{m+n}.

    21+(−9)+5=21−9+5=2−32^{1 + (-9) + 5} = 2^{1 - 9 + 5} = 2^{-3}

  5. Calculate the final value.

    2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}
    The simplified expression is 18\boxed{\frac{1}{8}}.

Part ii

181\frac{1}{81}

  1. Express all numbers in the expression as powers of 3.

    27=3327 = 3^3
    9=329 = 3^2
    81=3481 = 3^4
    Substitute these into the expression:

    (33)2/3×(32)5(34)4\frac{(3^3)^{2/3} \times (3^2)^5}{(3^4)^4}

  2. Apply the exponent rule (am)n=amn(a^m)^n = a^{mn} to each term.

    33×(2/3)×32×534×4=32×310316\frac{3^{3 \times (2/3)} \times 3^{2 \times 5}}{3^{4 \times 4}} = \frac{3^2 \times 3^{10}}{3^{16}}

  3. Apply the exponent rule am×an=am+na^m \times a^n = a^{m+n} to the numerator.

    32+10316=312316\frac{3^{2+10}}{3^{16}} = \frac{3^{12}}{3^{16}}

  4. Apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}.

    312−16=3−43^{12-16} = 3^{-4}

  5. Calculate the final value.

    3−4=134=1813^{-4} = \frac{1}{3^4} = \frac{1}{81}

    The simplified expression is 181\boxed{\frac{1}{81}}.

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