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
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(Type an integer or a simplified fraction.)
First, let's simplify the expression inside the parentheses.
StartFraction 8 minus left parenthesis negative 4 right parenthesis cubed Over 5 squared - 8 times 2 EndFraction
= StartFraction 8 - (-4)^3 Over 5^2 - 8 times 2 EndFraction
Next, let's simplify the exponent and the multiplication.
= StartFraction 8 - (-64) Over 25 - 8 times 2 EndFraction
= StartFraction 8 + 64 Over 25 - 16 EndFraction
= StartFraction 72 Over 25 - 16 EndFraction
= StartFraction 72 Over 25 - 16 times 25 Over 25 EndFraction
Next, let's find a common denominator and subtract.
= StartFraction 72 Over 25 - 400 Over 25 EndFraction
= StartFraction -328 Over 25 EndFraction
Therefore, the simplified expression is -328/25.
To evaluate the expression, let's break it down step by step:
Step 1: Evaluate the expression inside the parentheses:
Inside the parentheses, we have 8 minus (-4) which gives us 12.
Step 2: Evaluate the numerator of the fraction:
The numerator is (12^3), which means 12 multiplied by itself three times.
12^3 = 12 * 12 * 12 = 1,728.
Step 3: Evaluate the denominator of the fraction:
The denominator is (5^2 - 8 * 2), which means 5 squared minus 8 times 2.
5^2 = 5 * 5 = 25
8 * 2 = 16
So, the denominator is 25 - 16 = 9.
Step 4: Evaluate the fraction as a whole:
Now we have (1728 / 9).
Step 5: Square the fraction:
(1728 / 9)^2 = (1728^2) / (9^2)
= 2,985,984 / 81
So, the simplified expression is 2,985,984/81.