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Solve the following exponent problems: a) 60 + 60, b) 41 x 22 x 31, c) (84)/(2), d) (81 x 243)/(272 x 36)

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Answers

4 parts
a
2
b
48
c
2048
d

127\frac{1}{27}

Step-by-step solution

18 steps

Here are the solutions to the problems:

Part a· 2 steps

Determine the value of 60+606^0 + 6^0

  1. Apply the rule that any non-zero number raised to the power of 0 is 1.

    60=16^0 = 1

  2. Substitute the value and perform the addition.

    60+60=1+16^0 + 6^0 = 1 + 1 =2= 2

    The value is 2\boxed{2}.

Part b48

Determine the value of 41×22×314^1 \times 2^2 \times 3^1

  1. Evaluate each term.

    41=44^1 = 4 22=2×2=42^2 = 2 \times 2 = 4 31=33^1 = 3

  2. Multiply the evaluated terms.

    4×4×3=16×34 \times 4 \times 3 = 16 \times 3 =48= 48

    The value is 48\boxed{48}.

Part c2048

Determine the value of 842\frac{8^4}{2}

  1. Express the base 8 as a power of 2.

    8=238 = 2^3

  2. Substitute 232^3 for 8 in the expression.

    (23)42\frac{(2^3)^4}{2}

  3. Apply the exponent rule (am)n=am×n(a^m)^n = a^{m \times n}.

    23×42=21221\frac{2^{3 \times 4}}{2} = \frac{2^{12}}{2^1}

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

    212−1=2112^{12-1} = 2^{11}

  5. Calculate the value of 2112^{11}.

    211=20482^{11} = 2048

    The value is 2048\boxed{2048}.

Part d

127\frac{1}{27}

Determine the value of 81×243272×36\frac{81 \times 243}{27^2 \times 3^6}

  1. Express all numbers as powers of 3.

    81=3×3×3×3=3481 = 3 \times 3 \times 3 \times 3 = 3^4 243=3×3×3×3×3=35243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5 27=3×3×3=3327 = 3 \times 3 \times 3 = 3^3

  2. Substitute these into the expression.

    34×35(33)2×36\frac{3^4 \times 3^5}{(3^3)^2 \times 3^6}

  3. Apply the exponent rule (am)n=am×n(a^m)^n = a^{m \times n} to the denominator.

    (33)2=33×2=36(3^3)^2 = 3^{3 \times 2} = 3^6

  4. Rewrite the expression with all terms as powers of 3.

    34×3536×36\frac{3^4 \times 3^5}{3^6 \times 3^6}

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

    Numerator:34×35=34+5=39\text{Numerator:} \quad 3^4 \times 3^5 = 3^{4+5} = 3^9 Denominator:36×36=36+6=312\text{Denominator:} \quad 3^6 \times 3^6 = 3^{6+6} = 3^{12}

  6. Rewrite the expression.

    39312\frac{3^9}{3^{12}}

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

    39−12=3−33^{9-12} = 3^{-3}

  8. Apply the exponent rule a−n=1ana^{-n} = \frac{1}{a^n}.

    3−3=1333^{-3} = \frac{1}{3^3}

  9. Calculate the final value.

    13×3×3=127\frac{1}{3 \times 3 \times 3} = \frac{1}{27}

    The value is 127\boxed{\frac{1}{27}}.

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