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Chapter 5: Problem 115

Perform the indicated operations. Leave denominators in prime factorization form. \(\frac{1}{2^{4} \cdot 5^{3} \cdot 7}+\frac{1}{2 \cdot 5^{4}}-\frac{1}{2^{3} \cdot 5^{2}}\)

Short Answer

Expert verified
The simplified form of the expression \(\frac{1}{2^{4} \cdot 5^{3} \cdot 7}+\frac{1}{2 \cdot 5^{4}}-\frac{1}{2^{3} \cdot 5^{2}}\) is \(\frac{-9}{2^{4} \cdot 5^{4} \cdot 7}\).

Step by step solution

01

Identify the denominators

Given are three fractions \(\frac{1}{2^{4} \cdot 5^{3} \cdot 7}\),\(\frac{1}{2 \cdot 5^{4}}\) and \(\frac{1}{2^{3} \cdot 5^{2}}\). We need to identify each denominator and express it in prime factorization form.
02

Find the common denominator

To add or subtract fractions, we first need a common denominator. We can obtain it by finding the least common multiple (LCM) of the three denominators. In our case, LCM is \(2^{4} \cdot 5^{4}\cdot 7\).
03

Adjust the fractions to have common denominator

Now rewrite each fraction with the common denominator.For the first fraction, multiply the denominator by \(2^{0}\cdot 5^{1}\cdot 7^{0}\), and the numerator by the same.For the second, multiply the denominator by \(2^{3}\cdot 5^{0}\cdot 7^{1}\), and the numerator by the same.For the third fraction, multiply the denominator by \(2^{1}\cdot 5^{2}\cdot 7^{1}\), and the numerator by the same.Now, the fractions are\(\frac{1 \cdot 2^{0} \cdot 5^{1} \cdot 7^{0}}{2^{4} \cdot 5^{4} \cdot 7} + \frac{1 \cdot 2^{3} \cdot 5^{0} \cdot 7^{1}}{2^{4} \cdot 5^{4} \cdot 7} - \frac{1 \cdot 2^{1} \cdot 5^{2} \cdot 7^{1}}{2^{4} \cdot 5^{4} \cdot 7}\)
04

Add & subtract numerators

Now, perform the addition and subtraction operations. Add or subtract only the numerators and write the sum or difference over the common denominator.It becomes: \(\frac{5 + 56 - 70}{2^{4} \cdot 5^{4} \cdot 7}\).
05

Simplify the result

-9 is the result after performing the operations in the numerator. So, it simplifies to \(\frac{-9}{2^{4} \cdot 5^{4} \cdot 7}\).

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