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Chapter 6: Problem 501

Simplify. (a) \(3^{-4}\) (b) \(10^{-2}\)

Short Answer

Expert verified
Part (a): \frac{1}{81}, Part (b): \frac{1}{100}\.

Step by step solution

01

Understand Negative Exponents

Negative exponents indicate reciprocals. For any non-zero number a and positive integer n, the expression \(a^{-n} = \frac{1}{a^n}\).
02

Simplify part (a)

Using the rule from Step 1, rewrite \(3^{-4}\) as a reciprocal: \(3^{-4} = \frac{1}{3^4}\).
03

Calculate \(3^4\)

Evaluate the exponent in the denominator: \(3^4 = 81\). Thus, \(3^{-4} = \frac{1}{81}\).
04

Simplify part (b)

Similarly, rewrite \(10^{-2}\) as a reciprocal: \(10^{-2} = \frac{1}{10^2}\).
05

Calculate \(10^2\)

Evaluate the exponent in the denominator: \(10^2 = 100\). Thus, \(10^{-2} = \frac{1}{100}\).

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Key Concepts

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

Reciprocal
Negative exponents can be tricky, but understanding the concept of reciprocals makes it easier. When you see an exponent that is negative, it tells you to take the reciprocal of the number.
For example, if you have an expression like \(3^{-4}\), the negative sign means you will flip the base (in this case, 3) to make it a fraction.
This turns \(3^{-4}\) into \(\frac{1}{3^4}\).
The same goes for any number with a negative exponent. Another example is \(10^{-2}\). By taking the reciprocal, \(10^{-2}\) becomes \(\frac{1}{10^2}\).
This rule is consistent for any non-zero number with a negative exponent.
Simplifying Exponents
Simplifying exponents involves breaking down the problem into more manageable parts. When you have an expression like \(3^{-4}\), you first apply the reciprocal rule.
So, \(3^{-4}\) becomes \(\frac{1}{3^4}\).
Now you evaluate the exponent in the denominator.
So \(3^4\) means multiplying 3 by itself four times: \(3 \times 3 \times 3 \times 3 = 81\).
Thus, \(3^{-4}\) simplifies to \(\frac{1}{81}\).
Applying the same logic to another example, \(10^{-2}\), first rewrite it as \(\frac{1}{10^2}\).
Then, evaluate \(10^2\). That's 10 multiplied by itself twice, or \(10 \times 10 = 100\).
So, \(10^{-2}\) simplifies to \(\frac{1}{100}\).
Evaluating Exponents
Evaluating exponents means performing the operation the exponent indicates. For positive exponents, this is straightforward.
Let's say you have \(3^4\). This means you multiply 3 by itself four times: \(3 \times 3 \times 3 \times 3\).
When you do the math, you get 81.
For negative exponents, like \(3^{-4}\), you first take the reciprocal to get \(\frac{1}{3^4}\).
Then, you evaluate the positive exponent in the denominator, which we already know is 81.
Thus, \(3^{-4} = \frac{1}{81}\).
The same process applies for \(10^{-2}\). First, recognize that \(10^{-2}\) becomes \(\frac{1}{10^2}\).
Then, evaluate the exponent: \(10^2 = 100\).
Finally, \(10^{-2}\) is \(\frac{1}{100}\).

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