Evaluate: \frac{20}{5 \cdot 2^{-3}}
The Correct Answer and Explanation is:
The given expression is: 205⋅2−3\frac{20}{5 \cdot 2^{-3}}5⋅2−320
Step 1: Simplify the exponent
First, recall that 2−32^{-3}2−3 means: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}2−3=231=81
Now, substitute this value into the original expression: 205⋅18\frac{20}{5 \cdot \frac{1}{8}}5⋅8120
Step 2: Simplify the denominator
Next, simplify the denominator. When you multiply 5 by 18\frac{1}{8}81, you get: 5⋅18=585 \cdot \frac{1}{8} = \frac{5}{8}5⋅81=85
So, the expression becomes: 2058\frac{20}{\frac{5}{8}}8520
Step 3: Divide by a fraction
To divide by a fraction, you multiply by the reciprocal of the fraction. The reciprocal of 58\frac{5}{8}85 is 85\frac{8}{5}58. So the expression becomes: 20⋅8520 \cdot \frac{8}{5}20⋅58
Step 4: Simplify the multiplication
Now, multiply the numerator and denominator: 20⋅85=20⋅85=160520 \cdot \frac{8}{5} = \frac{20 \cdot 8}{5} = \frac{160}{5}20⋅58=520⋅8=5160
Now simplify: 1605=32\frac{160}{5} = 325160=32
Final Answer:
The value of the expression is 323232.
This step-by-step simplification shows how to handle negative exponents, multiplication, and division of fractions to reach the correct result.
