Problem 505 Simplify. (a) \(\frac{1}{c^{-5... [FREE SOLUTION]

Chapter 6: Problem 505

Simplify. (a) \(\frac{1}{c^{-5}}\) (b) \(\frac{1}{5^{-2}}\)

Short Answer

Expert verified

(a) \(c^{5}\), (b) 25

Step by step solution

Simplify the expression \(\frac{1}{c^{-5}}\)

To simplify \(\frac{1}{c^{-5}}\), use the property of exponents: \(a^{-n} = \frac{1}{a^n}\). This means \(\frac{1}{c^{-5}} = c^{5}\).

Simplify the expression \(\frac{1}{5^{-2}}\)

Similarly, use the property of exponents: \(a^{-n} = \frac{1}{a^n}\). This means \(\frac{1}{5^{-2}} = 5^{2}\).

Evaluate \(5^2\)

Now that the expression is simplified to \(5^{2}\), calculate \(5^{2} = 25\).

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Key Concepts

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

Properties of Exponents

When working with exponents, there are several fundamental properties that make it easier to simplify expressions. One of the most useful properties is that any base raised to a negative exponent can be rewritten as a fraction: \(a^{-n} = \frac{1}{a^n}\). This means the expression becomes a fraction where the exponent becomes positive and the base remains the same.
Another crucial property is that any number raised to the power of zero is one: \(a^0 = 1\). These properties help streamline calculations and make complex expressions more manageable.

Negative Exponents

Negative exponents might seem confusing at first, but they're quite simple once you understand the property of converting them into positive exponents. For example, \(a^{-n} = \frac{1}{a^n}\) tells us that a negative exponent means the reciprocal of the base raised to the positive exponent.
Imagine you have \(\frac{1}{c^{-5}}\). Using the property, replace the negative exponent by flipping it to the numerator as a positive exponent: \(\frac{1}{c^{-5}} = c^5\). This step is fundamental in simplifying expressions with negative exponents and should be memorized for efficiency.

Simplification

Simplification is about making an expression easier to work with. We apply the properties of exponents to rewrite expressions in a simpler form. For the given exercise, consider \(\frac{1}{5^{-2}}\).
First, use the property \(a^{-n} = \frac{1}{a^n}\) to convert the negative exponent to a positive one: \(\frac{1}{5^{-2}} = 5^2\). Then, evaluate the expression by calculating the power, \(5^2 = 25\).
When simplifying expressions with exponents, always remember to apply these properties, break down the steps, and simplify gradually.

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91影视

Simplify. (a) \(\frac{1}{c^{-5}}\) (b) \(\frac{1}{5^{-2}}\)

Short Answer

Step by step solution

Simplify the expression \(\frac{1}{c^{-5}}\)

Simplify the expression \(\frac{1}{5^{-2}}\)

Evaluate \(5^2\)

Key Concepts

Properties of Exponents

Negative Exponents

Simplification

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