Math

QuestionCalculate (23)24382\frac{\left(2^{3}\right)^{2}-4^{-3}}{8^{-2}}.

Studdy Solution

STEP 1

Assumptions1. The problem is to simplify the expression (3)438\frac{\left(^{3}\right)^{}-4^{-3}}{8^{-}}. . We will use the rules of exponents, which state that (am)n=amn(a^m)^n = a^{mn}, an=1ana^{-n} = \frac{1}{a^n}, and a0=1a^0 =1.
3. We will also use the fact that 4 =^ and 8=38 =^3 to simplify the expression.

STEP 2

First, we simplify the numerator of the fraction using the rule (am)n=amn(a^m)^n = a^{mn}.
(2)2=2×2=26\left(2^{}\right)^{2} =2^{ \times2} =2^6

STEP 3

Next, we simplify the second term in the numerator using the rule an=1ana^{-n} = \frac{1}{a^n} and the fact that =22 =2^2.
3=(22)3=22×3=26^{-3} = \left(2^{2}\right)^{-3} =2^{2 \times -3} =2^{-6}

STEP 4

Now, we simplify the denominator using the rule an=1ana^{-n} = \frac{1}{a^n} and the fact that 8=238 =2^3.
82=(23)2=23×2=268^{-2} = \left(2^{3}\right)^{-2} =2^{3 \times -2} =2^{-6}

STEP 5

We substitute the simplified expressions from steps2,3, and4 into the original expression.
(23)24382=222\frac{\left(2^{3}\right)^{2}-4^{-3}}{8^{-2}} = \frac{2^ -2^{-}}{2^{-}}

STEP 6

We can rewrite the fraction as a division operation.
262626=26÷2626÷26\frac{2^6 -2^{-6}}{2^{-6}} =2^6 \div2^{-6} -2^{-6} \div2^{-6}

STEP 7

We simplify the division operations using the rule am÷an=amna^m \div a^n = a^{m-n}.
26÷2626÷26=26(6)26(6)=212202^6 \div2^{-6} -2^{-6} \div2^{-6} =2^{6 - (-6)} -2^{-6 - (-6)} =2^{12} -2^0

STEP 8

We simplify 20=12^0 =1.
21220=21212^{12} -2^0 =2^{12} -1

STEP 9

Finally, we calculate the value of 2122^{12} and subtract.
212=4096=40952^{12} - =4096 - =4095So, (23)24382=4095\frac{\left(2^{3}\right)^{2}-4^{-3}}{8^{-2}} =4095.

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