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Chapter 5: Problem 106

Perform the indicated operations. Express each answer as a fraction reduced to its lowest terms. \(\frac{5^{6}}{5^{4}}-\frac{2^{4}}{2^{6}}\)

Short Answer

Expert verified
The simplified expression is \(\frac{99}{4}\).

Step by step solution

01

Applying exponent properties

For the expression \(\frac{5^{6}}{5^{4}}\), apply the rule of the exponents that says \(a^{m}/a^{n}=a^{(m-n)}\). Therefore, it simplifies to \(5^{(6-4)}\) or \(5^{2}\). Similarly, \(\frac{2^{4}}{2^{6}}\) simplifies to \(2^{(4-6)}\) or \(2^{-2}\).
02

Simplifying the expression

Next, simplify \(5^2\) and \(2^{-2}\). It is well known that \(5^{2}=25\). For \(-2^{-2}\), because the exponent is negative, we put it in the denominator, so it becomes \(\frac{1}{2^{2}}\), which simplifies to \(1/4\).
03

Performing the operations

Subtract the fraction \(\frac{1}{4}\) from the number 25 to get the final result: \(25-1/4=25-(1/4)\). Taking \(4\) as the common denominator, the expression becomes \(\frac{100}{4}-\frac{1}{4}=\frac{99}{4}\).

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