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

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Step 1: Write the sum explicitly. S = ∑_(k=1)¹⁰ (2)/(3^k) = 2/3 + (2)/(3²) + (2)/(3³) + + (2)/(3¹⁰) Step 2: Factor out the constant 2. S = 2 ∑_(k=1)¹⁰ ( 1/3 )^k; Step 3: Recall the formula for the sum of a finite geometric series ∑_(k=1)ⁿ r^k. ∑_(k=1)ⁿ r^k = r (1 − rⁿ)/(1 − r); Step 4: Substitute r = 1/3 and n = 10. ∑_(k=1)¹⁰ ( 1/3 )^k = (1/3 ( 1 − ( 1/3 )¹⁰ ))/(1 − 1/3) Step 5: Simplify the denominator. 1 − 1/3 = 2/3; ∑_(k=1)¹⁰ ( 1/3 )^k = (1/3 ( 1 − 3⁻¹⁰ ))/(2/3) = 1/3 · 3/2 ( 1 − 3⁻¹⁰ ) = 1/2 ( 1 − 3⁻¹⁰ ) Step 6: Multiply by 2 to find S. S = 2 · 1/2 ( 1 − 3⁻¹⁰ ) = 1 − 3⁻¹⁰ = 1 − (1)/(3¹⁰) Step 7: Compute 3¹⁰ step by step. 3¹ = 3; 3² = 9; 3³ = 27; 3⁴ = 81; 3⁵ = 243; 3⁶ = 729; 3⁷ = 2187; 3⁸ = 6561; 3⁹ = 19683; 3¹⁰ = 59049 Step 8: Substitute the value. S = 1 − 1/59049 = (59049 − 1)/(59049) = 59048/59049 Final answer: 59048/59049

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

Answer

Step-by-step solution

Write the sum explicitly.

Factor out the constant 2.

Recall the formula for the sum of a finite geometric series $\sum_{k=1}^n r^k$ .

Substitute $r = \frac{1}{3}$ and $n = 10$ .

Simplify the denominator.

Multiply by 2 to find $S$ .

Compute $3^{10}$ step by step.

Substitute the value.

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