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Evaluate the equation

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6 parts
a

−114-\frac{1}{14}

b

33x−23^{3x-2}

c

6285\frac{628}{5}

d

32n+33^{2n+3}

e

50343\frac{50}{343}

f

1310721125\frac{131072}{1125}

Step-by-step solution

20 steps

Here are the evaluations for each expression:

Part a· 2 steps

Evaluate 2m−1−2m−22m−1−2m+2\frac{2^{m-1} - 2^{m-2}}{2^{m-1} - 2^{m+2}}

  1. Factor out the smallest power of 2 from the numerator and denominator.

    Numerator: 2m−2(2(m−1)−(m−2)−1)=2m−2(21−1)=2m−2(1)=2m−22^{m-2}(2^{(m-1)-(m-2)} - 1) = 2^{m-2}(2^1 - 1) = 2^{m-2}(1) = 2^{m-2}
    Denominator: 2m−1(1−2(m+2)−(m−1))=2m−1(1−2m+2−m+1)=2m−1(1−23)=2m−1(1−8)=2m−1(−7)2^{m-1}(1 - 2^{(m+2)-(m-1)}) = 2^{m-1}(1 - 2^{m+2-m+1}) = 2^{m-1}(1 - 2^3) = 2^{m-1}(1 - 8) = 2^{m-1}(-7)

  2. Simplify the expression.

    2m−2−7⋅2m−1=−17⋅2(m−2)−(m−1)=−17⋅2−1=−17⋅12=−114\frac{2^{m-2}}{-7 \cdot 2^{m-1}} = -\frac{1}{7} \cdot 2^{(m-2)-(m-1)} = -\frac{1}{7} \cdot 2^{-1} = -\frac{1}{7} \cdot \frac{1}{2} = -\frac{1}{14}

    The evaluated expression is −114\boxed{-\frac{1}{14}}.

Part b

33x−23^{3x-2}

Evaluate 310+x×273x−437x\frac{3^{10+x} \times 27^{3x-4}}{3^{7x}}

  1. Express all terms with base 3.

    273x−4=(33)3x−4=33(3x−4)=39x−1227^{3x-4} = (3^3)^{3x-4} = 3^{3(3x-4)} = 3^{9x-12}

  2. Substitute and apply exponent rules for multiplication and division.

    310+x×39x−1237x=3(10+x)+(9x−12)37x=310x−237x\frac{3^{10+x} \times 3^{9x-12}}{3^{7x}} = \frac{3^{(10+x) + (9x-12)}}{3^{7x}} = \frac{3^{10x-2}}{3^{7x}}

  3. Simplify the expression.

    3(10x−2)−7x=33x−23^{(10x-2) - 7x} = 3^{3x-2}

    The evaluated expression is 33x−2\boxed{3^{3x-2}}.

Part c

6285\frac{628}{5}

Evaluate 3×125r+1+9×53r−18×53r−5×125r\frac{3 \times 125^{r+1} + 9 \times 5^{3r-1}}{8 \times 5^{3r} - 5 \times 125^r}

  1. Express all terms with base 5.

    125r+1=(53)r+1=53r+3125^{r+1} = (5^3)^{r+1} = 5^{3r+3} 125r=(53)r=53r125^r = (5^3)^r = 5^{3r}

  2. Substitute and factor out common terms.

    3×53r+3+9×53r−18×53r−5×53r=3×53r×53+9×53r×5−153r(8−5)\frac{3 \times 5^{3r+3} + 9 \times 5^{3r-1}}{8 \times 5^{3r} - 5 \times 5^{3r}} = \frac{3 \times 5^{3r} \times 5^3 + 9 \times 5^{3r} \times 5^{-1}}{5^{3r}(8 - 5)}

  3. Simplify the numerator and denominator.

    Numerator: 53r(3×125+9×15)=53r(375+95)=53r(1875+95)=53r188455^{3r}(3 \times 125 + 9 \times \frac{1}{5}) = 5^{3r}(375 + \frac{9}{5}) = 5^{3r}(\frac{1875+9}{5}) = 5^{3r} \frac{1884}{5}
    Denominator: 53r(3)5^{3r}(3)

  4. Simplify the fraction.

    53r1884553r×3=18845×3=188415\frac{5^{3r} \frac{1884}{5}}{5^{3r} \times 3} = \frac{1884}{5 \times 3} = \frac{1884}{15}

  5. Reduce the fraction.

    1884÷315÷3=6285\frac{1884 \div 3}{15 \div 3} = \frac{628}{5}

    The evaluated expression is 6285\boxed{\frac{628}{5}}.

Part d

32n+33^{2n+3}

Evaluate 273n×35×32n9×311n×3−2n\frac{27^{3n} \times 3^5 \times 3^{2n}}{9 \times 3^{11n} \times 3^{-2n}}

  1. Express all terms with base 3.

    273n=(33)3n=39n27^{3n} = (3^3)^{3n} = 3^{9n} 9=329 = 3^2

  2. Substitute and apply exponent rules for multiplication.

    39n×35×32n32×311n×3−2n=39n+5+2n32+11n−2n=311n+539n+2\frac{3^{9n} \times 3^5 \times 3^{2n}}{3^2 \times 3^{11n} \times 3^{-2n}} = \frac{3^{9n+5+2n}}{3^{2+11n-2n}} = \frac{3^{11n+5}}{3^{9n+2}}

  3. Apply exponent rule for division.

    3(11n+5)−(9n+2)=311n+5−9n−2=32n+33^{(11n+5) - (9n+2)} = 3^{11n+5-9n-2} = 3^{2n+3}

    The evaluated expression is 32n+3\boxed{3^{2n+3}}.

Part e

50343\frac{50}{343}

Evaluate 7n−2+7n3×7n+4×7n\frac{7^{n-2} + 7^n}{3 \times 7^n + 4 \times 7^n}

  1. Factor out common terms from the numerator and denominator.

    Numerator: 7n−2+7n=7n⋅7−2+7n⋅1=7n(7−2+1)=7n(149+1)=7n(1+4949)=7n50497^{n-2} + 7^n = 7^n \cdot 7^{-2} + 7^n \cdot 1 = 7^n(7^{-2} + 1) = 7^n(\frac{1}{49} + 1) = 7^n(\frac{1+49}{49}) = 7^n \frac{50}{49}
    Denominator: 3×7n+4×7n=7n(3+4)=7n(7)3 \times 7^n + 4 \times 7^n = 7^n(3+4) = 7^n(7)

  2. Simplify the fraction.

    7n50497n⋅7=50497=5049×7=50343\frac{7^n \frac{50}{49}}{7^n \cdot 7} = \frac{\frac{50}{49}}{7} = \frac{50}{49 \times 7} = \frac{50}{343}

    The evaluated expression is 50343\boxed{\frac{50}{343}}.

Part f

1310721125\frac{131072}{1125}

Evaluate 45×64×123303×36\frac{4^5 \times 6^4 \times 12^3}{30^3 \times 3^6}

  1. Express all bases as products of prime factors.

    4=224 = 2^2 6=2×36 = 2 \times 3 12=22×312 = 2^2 \times 3 30=2×3×530 = 2 \times 3 \times 5

  2. Substitute prime factorizations into the expression.

    (22)5×(2×3)4×(22×3)3(2×3×5)3×36=210×24×34×26×3323×33×53×36\frac{(2^2)^5 \times (2 \times 3)^4 \times (2^2 \times 3)^3}{(2 \times 3 \times 5)^3 \times 3^6} = \frac{2^{10} \times 2^4 \times 3^4 \times 2^6 \times 3^3}{2^3 \times 3^3 \times 5^3 \times 3^6}

  3. Combine terms with the same base in the numerator and denominator.

    Numerator: 210+4+6×34+3=220×372^{10+4+6} \times 3^{4+3} = 2^{20} \times 3^7
    Denominator: 23×33+6×53=23×39×532^3 \times 3^{3+6} \times 5^3 = 2^3 \times 3^9 \times 5^3

  4. Apply exponent rules for division.

    220×3723×39×53=220−3×37−9×50−3=217×3−2×5−3\frac{2^{20} \times 3^7}{2^3 \times 3^9 \times 5^3} = 2^{20-3} \times 3^{7-9} \times 5^{0-3} = 2^{17} \times 3^{-2} \times 5^{-3}

  5. Write the expression with positive exponents and calculate the final value.

    21732×53=1310729×125=1310721125\frac{2^{17}}{3^2 \times 5^3} = \frac{131072}{9 \times 125} = \frac{131072}{1125}

    The evaluated expression is 1310721125\boxed{\frac{131072}{1125}}.

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