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Chapter 4: Problem 53

\(\frac{5}{5^{-1}}\)

Short Answer

Expert verified
25

Step by step solution

01

Understand the Problem

The given problem involves a fraction where the denominator is an exponent. We need to simplify the expression \(\frac{5}{5^{-1}}\).
02

Apply the Reciprocal Property

Recall that \(5^{-1} = \frac{1}{5}\). This means the expression becomes \(\frac{5}{\frac{1}{5}}\).
03

Simplify the Complex Fraction

To simplify \(\frac{5}{\frac{1}{5}}\), multiply 5 by the reciprocal of \(\frac{1}{5}\), which is 5. Hence, \(\frac{5}{\frac{1}{5}} = 5 \times 5 = 25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal Property
When simplifying fractions, understanding the reciprocal property is vital. A reciprocal of a number is essentially its 'flip.' For example, the reciprocal of \(5\) is \(\frac{1}{5}\).
In our exercise, we had \(5^{-1}\). According to the reciprocal property, \(5^{-1}\) is the same as \(\frac{1}{5}\).
Thus, rewriting the expression using this property simplifies our work significantly. Always remember: \(a^{-1} = \frac{1}{a}\). This will help you tackle various problems involving negative exponents and fractions.
Negative Exponents
Negative exponents can often be confusing, but they're quite simple once you get the hang of them. Essentially, a negative exponent indicates that the number should be taken as a reciprocal.
For instance, \(x^{-1}\) is the reciprocal of \(x\), which is \(\frac{1}{x}\).
So, whenever you see a negative exponent, think 'flip the number.'
Let's go back to our exercise: We had \(5^{-1}\), which equals \(\frac{1}{5}\). Understanding this concept helps you break down and solve expressions involving negative exponents.
Complex Fractions
Complex fractions might seem daunting at first, but breaking them down step by step makes them easier to handle. A complex fraction is a fraction where either the numerator, the denominator, or both, are also fractions.
In our exercise, \(\frac{5}{5^{-1}}\) translates to \(\frac{5}{\frac{1}{5}}\). This can be intimidating at first sight, but simplifying it involves a straightforward step: multiplying by the reciprocal of the denominator.
So, \(\frac{5}{\frac{1}{5}}\) becomes \(5 \times 5 = 25\). By understanding and applying this rule, you can simplify even the most complex-looking fractions.

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Most popular questions from this chapter

Find each product. $$ (5 a+3 b)(5 a-4 b) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$

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