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Simplify the term (2ⁿ)³ using the exponent rule (a^m)ⁿ = a^mn.

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3 parts
e

26n2^{6n}

f

32m3^{2m}

g

a154a^{\frac{15}{4}}

Step-by-step solution

12 steps

Here are the solutions for the expressions:

Part e· 3 steps

Simplify 2n×(2n)3×22n2^n \times (2^n)^3 \times 2^{2n}

  1. Simplify the term (2n)3(2^n)^3 using the exponent rule (am)n=amn(a^m)^n = a^{mn}.

    (2n)3=2n×3=23n(2^n)^3 = 2^{n \times 3} = 2^{3n}

  2. Substitute this back into the expression.

    2n×23n×22n2^n \times 2^{3n} \times 2^{2n}

  3. Combine the terms using the exponent rule am×an×ap=am+n+pa^m \times a^n \times a^p = a^{m+n+p}.

    2n×23n×22n=2n+3n+2n2^n \times 2^{3n} \times 2^{2n} = 2^{n + 3n + 2n} 2n+3n+2n=26n2^{n + 3n + 2n} = 2^{6n}

    The simplified expression is 26n\boxed{2^{6n}}.

Part f

32m3^{2m}

Simplify 3m×27m×9−m3^m \times 27^m \times 9^{-m}

  1. Express all bases as powers of 3.

    27=3327 = 3^3 9=329 = 3^2

  2. Substitute these into the expression.

    3m×(33)m×(32)−m3^m \times (3^3)^m \times (3^2)^{-m}

  3. Simplify the terms with powers of powers using (ax)y=axy(a^x)^y = a^{xy}.

    (33)m=33m(3^3)^m = 3^{3m} (32)−m=32×(−m)=3−2m(3^2)^{-m} = 3^{2 \times (-m)} = 3^{-2m}

  4. Substitute these back into the expression.

    3m×33m×3−2m3^m \times 3^{3m} \times 3^{-2m}

  5. Combine the terms using the exponent rule ax×ay×az=ax+y+za^x \times a^y \times a^z = a^{x+y+z}.

    3m×33m×3−2m=3m+3m+(−2m)3^m \times 3^{3m} \times 3^{-2m} = 3^{m + 3m + (-2m)} 3m+3m−2m=34m−2m3^{m + 3m - 2m} = 3^{4m - 2m} 34m−2m=32m3^{4m - 2m} = 3^{2m}

    The simplified expression is 32m\boxed{3^{2m}}.

Part g

a154a^{\frac{15}{4}}

Simplify (a1/2×a)2.5(a^{1/2} \times a)^{2.5}

  1. Simplify inside the parenthesis.

    Remember that a=a1a = a^1. Use the exponent rule ax×ay=ax+ya^x \times a^y = a^{x+y}.

    a1/2×a1=a1/2+1a^{1/2} \times a^1 = a^{1/2 + 1} a1/2+1=a1/2+2/2=a3/2a^{1/2 + 1} = a^{1/2 + 2/2} = a^{3/2}

  2. Substitute this back into the expression.

    (a3/2)2.5(a^{3/2})^{2.5}

  3. Convert the decimal exponent 2.5 to a fraction.

    2.5=2510=522.5 = \frac{25}{10} = \frac{5}{2}

  4. Apply the outer exponent using the rule (ax)y=axy(a^x)^y = a^{xy}.

    (a3/2)5/2=a32×52(a^{3/2})^{5/2} = a^{\frac{3}{2} \times \frac{5}{2}} a32×52=a3×52×2a^{\frac{3}{2} \times \frac{5}{2}} = a^{\frac{3 \times 5}{2 \times 2}} a154a^{\frac{15}{4}}

    The simplified expression is a154\boxed{a^{\frac{15}{4}}}.

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