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Chapter 7: Problem 2

Simplify each of the following exponential expressions. $$ \left(5^{3}\right)^{-1} $$

Short Answer

Expert verified
\( \frac{1}{125} \)

Step by step solution

01

Understand the negative exponent rule

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

Apply the negative exponent rule

Using the rule from Step 1, apply it to \( (5^3)^{-1} \). This gives us: \( (5^3)^{-1} = \frac{1}{5^3} \).
03

Simplify the expression

Now, calculate the value of \( 5^3 \) which is \( 5 \times 5 \times 5 = 125 \). Thus, the expression simplifies to \( \frac{1}{125} \).

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Key Concepts

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

Negative Exponent Rule
When you encounter a negative exponent in an expression, it can seem a little confusing at first. But don't worry鈥攊t's quite simple once you get the hang of it. A negative exponent tells you that you need to take the reciprocal of the base number and then raise it to the opposite (positive) exponent.
For example, if you see something like \(a^{-n}\), you can rewrite this as \( \frac{1}{a^n} \).
So, a negative exponent essentially 'flips' the base number to its reciprocal. This helps simplify the expression and makes it easier to handle.
Using this rule, in the exercise, we turned \( (5^3)^{-1} \) into \( \frac{1}{5^3} \). Simple, right?
Reciprocals
The term 'reciprocal' might sound technical, but it's actually quite simple. A reciprocal of a number is just 1 divided by that number.
For instance, the reciprocal of 5 is \( \frac{1}{5} \). Now, if we have an exponent attached to that base (like \( 5^3 \)), it works the same way.
If you need the reciprocal of a larger number like \( 5^3 \), it's \( \frac{1}{5^3} \). So, when we turned \( (5^3)^{-1} \) into \( \frac{1}{5^3} \), we were essentially finding the reciprocal of \( 5^3 \).
This helps in simplifying complex exponential expressions, especially when negative exponents are involved.
Powers of Numbers
Understanding powers (or exponents) of numbers is a key concept in simplifying exponential expressions.
When you see \( 5^3 \), this simply means multiplying 5 by itself three times:
\( 5 \times 5 \times 5 \).
This gives you 125. Powers help in condensing repeated multiplication into a more manageable form.
In our exercise, after using the negative exponent rule to move the exponent to the denominator, we calculated \( 5^3 \) as 125, thus simplifying \( \frac{1}{5^3} \) to \( \frac{1}{125} \). Mastering the concept of powers and exponents will make it easier for you to handle and simplify similar expressions in the future.

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Most popular questions from this chapter

Compute the value of each of the following exponential expressions. $$ 3(5)^{2} $$

Compute the value of each of the following exponential expressions. $$ 3^{2}+3^{1} $$

Simplify each of the following exponential expressions. $$ \left(5^{-2}\right)^{-2} $$

Simplify the following expressions. $$ \frac{2^{4}}{2^{-2}} $$

Find the reciprocal of each of the following numbers. $$ 3 $$
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