Evaluate the following expression.
left parenthesis StartFraction 8 minus left parenthesis negative 4 right parenthesis cubed Over 5 squared minus 8 times 2 EndFraction right parenthesis squared
First, let's simplify the innermost parentheses:
StartFraction 8 - ( -4 ) EndFraction = StartFraction 8 + 4 EndFraction = 12
Next, we'll evaluate the exponent:
(12)^3 = 12 × 12 × 12 = 1,728
Now let's simplify the fraction:
StartFraction 1,728 Over 5^2 - 8 × 2 EndFraction
StartFraction 1,728 Over 25 - 8 × 2 EndFraction = StartFraction 1,728 Over 25 - 16 EndFraction
Next, let's calculate the values:
StartFraction 1,728 Over 25 - 16 EndFraction = StartFraction 1,728 Over 25 -16 EndFraction = StartFraction 1,728 - 400 Over 25 EndFraction = StartFraction 1,328 Over 25 EndFraction
Finally, let's square this fraction:
StartFraction 1,328 Over 25 EndFraction^2 = StartFraction 1,328^2 Over 25^2 EndFraction = StartFraction 1,764,384 Over 625 EndFraction
The simplified expression is 1,764,384/625.
To evaluate the given expression:
1. First, we evaluate the expression inside the parentheses:
StartFraction 8 minus left parenthesis negative 4 right parenthesis cubed Over 5 squared minus 8 times 2 EndFraction.
2. Evaluate the expression inside the numerator of the fraction:
The cube of (-4) is equal to (-4) × (-4) × (-4), which is equal to -64.
So, the numerator becomes -64.
3. Evaluate the expression inside the denominator of the fraction:
The square of 5 is 5 × 5, which is equal to 25.
The result of 8 times 2 is 16.
So, the denominator becomes 25 - 16, which is equal to 9.
4. Replace the fraction in the original expression with the simplified fraction from step 3:
left parenthesis -64 Over 9 right parenthesis squared.
5. Evaluate the expression inside the parentheses:
The square of -64 is (-64) × (-64), which is equal to 4096.
So, the expression becomes 4096.
Therefore, the original expression
left parenthesis StartFraction 8 minus left parenthesis negative 4 right parenthesis cubed Over 5 squared minus 8 times 2 EndFraction right parenthesis squared
evaluates to 4096.