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Chapter 4: Problem 27

Simplify. Do not leave negative exponents in your answer. $$ 10^{-2} $$

Short Answer

Expert verified
\(10^{-2} = \frac{1}{100}

Step by step solution

01

Understand Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. For example, for any non-zero number a and positive integer n, the expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).
02

Rewrite the Expression with Positive Exponents

Apply the rule from Step 1 to rewrite \(10^{-2}\). This gives us \(\frac{1}{10^2}\).
03

Simplify the Expression

Evaluate the expression with the positive exponent: \(10^2 = 100\). Thus, \(\frac{1}{10^2} \) simplifies to \ \frac{1}{100}\.

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Key Concepts

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

reciprocal
The concept of the reciprocal is key when working with negative exponents. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of 5 is \frac{1}{5}\. This is because 5 multiplied by \frac{1}{5}\ equals 1, which is the defining feature of reciprocals.
  • For any non-zero number \(a\), the reciprocal is \frac{1}{a}\.
  • This principle helps in understanding negative exponents.
When you see a negative exponent, say \(a^{-n}\), it means you need to take the reciprocal of the number first and then raise it to the positive power. For instance, \(10^{-2}\) is the same as \frac{1}{10^{2}\.
positive exponents
Positive exponents are straightforward. An exponent expresses how many times to multiply a number by itself. For example, \(10^2\) means 10 multiplied by 10, which equals 100.
If you see \(a^n\), all you have to do is multiply \(a\) by itself \(n\) times.
  • \(2^3 = 2 \times 2 \times 2 = 8\)
  • \(5^4 = 5 \times 5 \times 5 \times 5 = 625\)
By rewriting negative exponents using positive exponents, you simplify complex expressions. For example, changing \(10^{-2}\) to \frac{1}{10^2}\ is the first step to make the exponent positive.
simplifying expressions
Simplifying expressions is the process of making equations easier to work with. Once you convert negative exponents to positive ones and understand how to work with positive exponents, you are ready to simplify the expression.
  • First, change the negative exponent to a reciprocal: \(10^{-2}\) becomes \frac{1}{10^2}\.
  • Next, evaluate the positive exponent: \(10^2 = 100\).
  • Finally, simplify the fraction: \frac{1}{100}\.
This method reduces the complexity and makes it easier to handle calculations, ensuring you're not left with negative exponents in your final answer.

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