Simplify: 2(x)2 x^(1/4) xy8 / (3x)0 * (3x4)

Mathematics
Simplify: 2(x)2 x^(1/4) xy8 / (3x)0 * (3x4)

ScanSolve AI Answer

Answer

5904859049\dfrac{59048}{59049}

Step 1: Write the sum explicitly.

S=∑k=11023k=23+232+233+⋯+2310S = \sum_{k=1}^{10} \frac{2}{3^k} = \frac{2}{3} + \frac{2}{3^2} + \frac{2}{3^3} + \cdots + \frac{2}{3^{10}}

Step 2: Factor out the constant 2.

S=2∑k=110(13)kS = 2 \sum_{k=1}^{10} \left( \frac{1}{3} \right)^k

Step 3: Recall the formula for the sum of a finite geometric series ∑k=1nrk\sum_{k=1}^n r^k.

∑k=1nrk=r1−rn1−r\sum_{k=1}^n r^k = r \frac{1 - r^n}{1 - r}

Step 4: Substitute r=13r = \frac{1}{3} and n=10n = 10.

∑k=110(13)k=13(1−(13)10)1−13\sum_{k=1}^{10} \left( \frac{1}{3} \right)^k = \frac{\frac{1}{3} \left( 1 - \left( \frac{1}{3} \right)^{10} \right)}{1 - \frac{1}{3}}

Step 5: Simplify the denominator.

1−13=231 - \frac{1}{3} = \frac{2}{3} ∑k=110(13)k=13(1−3−10)23=13⋅32(1−3−10)=12(1−3−10)\sum_{k=1}^{10} \left( \frac{1}{3} \right)^k = \frac{\frac{1}{3} \left( 1 - 3^{-10} \right)}{\frac{2}{3}} = \frac{1}{3} \cdot \frac{3}{2} \left( 1 - 3^{-10} \right) = \frac{1}{2} \left( 1 - 3^{-10} \right)

Step 6: Multiply by 2 to find SS.

S=2⋅12(1−3−10)=1−3−10=1−1310S = 2 \cdot \frac{1}{2} \left( 1 - 3^{-10} \right) = 1 - 3^{-10} = 1 - \frac{1}{3^{10}}

Step 7: Compute 3103^{10} step by step.

31=33^1 = 3 32=93^2 = 9 33=273^3 = 27 34=813^4 = 81 35=2433^5 = 243 36=7293^6 = 729 37=21873^7 = 2187 38=65613^8 = 6561 39=196833^9 = 19683 310=590493^{10} = 59049

Step 8: Substitute the value.

S=1−159049=59049−159049=5904859049S = 1 - \frac{1}{59049} = \frac{59049 - 1}{59049} = \frac{59048}{59049}

Final answer: 5904859049\boxed{\dfrac{59048}{59049}}

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