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Chapter 5: Problem 11

Express using positive exponents and, if possible, simplify. $$7^{-2}$$

Short Answer

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Step by step solution

01

- Understand Negative Exponents

A negative exponent can be converted to a positive exponent by taking the reciprocal of the base. For example, for any nonzero number 'a' and a negative exponent '-n',
02

- Apply the Reciprocal Rule

Using the rule from Step 1, convert the negative exponent
03

- Simplify the Expression

Once we have the positive exponent, we can rewrite the expression more simply. Therefore,

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Key Concepts

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

Exponent Rules
Understanding exponent rules is key to solving many algebraic problems. An exponent indicates how many times a base number is multiplied by itself. When dealing with negative exponents, like in the given problem \(7^{-2}\), the rules change slightly. A negative exponent means you take the reciprocal of the base number and then apply the positive exponent. This doesn鈥檛 change the value, just the notation. For example, \(a^{-n} = \frac{1}{a^n}\). These rules help in converting and simplifying expressions into forms that are easier to handle.
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form. Let鈥檚 work through the provided exercise: \(7^{-2}\). Firstly, we use the rule we鈥檝e just discussed: a negative exponent tells us to take the reciprocal.\
So, \(7^{-2}\) becomes \(\frac{1}{7^2}\). Now, we deal with the positive exponent. Calculating \(7^2\) gives us \(49\). Therefore, \(\frac{1}{49}\). This is much simpler and easier to understand compared to the original expression. Simplifying expressions is like cleaning up math problems to make them neat and tidy.
Reciprocal
The concept of a reciprocal is vital when dealing with negative exponents. The reciprocal of a number is 1 divided by that number. For example, the reciprocal of \(a\) is \(\frac{1}{a}\). In our problem, the base \(7\) with a negative exponent becomes its reciprocal when converted to a positive exponent. So, \(7^{-2}\) becomes \(\frac{1}{7^2}\). Working with reciprocals transforms tricky negative exponents into manageable positive ones, simplifying the overall calculation. Remember, to find the reciprocal, just flip the number over to make it a fraction.

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Most popular questions from this chapter

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