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Rewrite the terms in the numerator and denominator using exponent rules a^m+n = a^m * aⁿ.

Mathematics

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Answers

2 parts
a

43\frac{4}{3}

b

14\frac{1}{4}

Step-by-step solution

8 steps

Part a· 4 steps

  1. Rewrite the terms in the numerator and denominator using exponent rules am+n=am⋅ana^{m+n} = a^m \cdot a^n.

    3n+1+9⋅3n6⋅3n+3n+1=3n⋅31+9⋅3n6⋅3n+3n⋅31\frac{3^{n+1} + 9 \cdot 3^n}{6 \cdot 3^n + 3^{n+1}} = \frac{3^n \cdot 3^1 + 9 \cdot 3^n}{6 \cdot 3^n + 3^n \cdot 3^1}

  2. Factor out the common term 3n3^n from the numerator and the denominator.

    3n(3+9)3n(6+3)\frac{3^n(3 + 9)}{3^n(6 + 3)}

  3. Simplify the expression by canceling out 3n3^n and performing the additions.

    3+96+3=129\frac{3 + 9}{6 + 3} = \frac{12}{9}

  4. Reduce the fraction to its simplest form.

    129=43\frac{12}{9} = \frac{4}{3}

    The simplified expression is 43\boxed{\frac{4}{3}}.

Part b

14\frac{1}{4}

  1. Rewrite the terms using exponent rules am+n=am⋅ana^{m+n} = a^m \cdot a^n and axy=(ax)ya^{xy} = (a^x)^y.

    Note that 25x=(52)x=52x25^x = (5^2)^x = 5^{2x}.

    52x+1−25x3⋅52x+1+25x=52x⋅51−52x3⋅52x⋅51+52x\frac{5^{2x+1} - 25^x}{3 \cdot 5^{2x+1} + 25^x} = \frac{5^{2x} \cdot 5^1 - 5^{2x}}{3 \cdot 5^{2x} \cdot 5^1 + 5^{2x}}

  2. Factor out the common term 52x5^{2x} from the numerator and the denominator.

    52x(5−1)52x(3⋅5+1)\frac{5^{2x}(5 - 1)}{5^{2x}(3 \cdot 5 + 1)}

  3. Simplify the expression by canceling out 52x5^{2x} and performing the arithmetic.

    5−115+1=416\frac{5 - 1}{15 + 1} = \frac{4}{16}

  4. Reduce the fraction to its simplest form.

    416=14\frac{4}{16} = \frac{1}{4}

    The simplified expression is 14\boxed{\frac{1}{4}}.

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