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Chapter 10: Problem 34

Simplify. Write the answer with positive exponents only. $$8^{-1}$$

Short Answer

Expert verified
\frac{1}{8}

Step by step solution

01

Identify the Given Expression

The problem provides the expression \(8^{-1}\).
02

Understand Negative Exponents

Negative exponents indicate that the base is on the wrong side of the fraction line. A negative exponent means you divide by that number of factors instead of multiplying by them.
03

Rewrite Using Positive Exponent

Rewrite \(8^{-1}\) by moving the base to the denominator: \[8^{-1} = \frac{1}{8^{1}}\]
04

Simplify the Expression

Since \(8^{1} = 8\), the simplified expression is \(\frac{1}{8}\).

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Key Concepts

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

Simplifying Expressions
Understanding how to simplify expressions is a crucial skill in mathematics. Simplifying means making the expression as basic as possible. For instance, consider the expression \(8^{-1}\).
It looks a bit complex but can be simplified by applying the rules of exponents. Always look for opportunities to rewrite expressions in a more straightforward form. This not only makes calculations easier but also helps to understand the underlying mathematics better. Breaking down each step, like we do in simpler problems, will become vital in more complex scenarios later. Remember, practice is key!
Positive Exponents
Exponents are a shorthand way of expressing repeated multiplication. When an exponent is positive, it tells us that we are multiplying the base by itself a certain number of times. For example, \(8^{1}\) simply means 8, since we are only 'multiplying' 8 once.
Negative exponents, on the other hand, indicate division rather than multiplication. Converting negative exponents to positive ones involves rewriting the expression to reflect division. For example, \(8^{-1} = \frac{1}{8}\). Such conversion simplifies the task of handling the expression.
Fractional Representation
Fractional representation is another way of expressing division. When we have a negative exponent like \(8^{-1}\), it's equivalent to writing \(\frac{1}{8}\).
This method of representing expressions allows us to see them from a different angle, often making it easier to understand and simplify.
So, when you see \(8^{-1}\), think of it as \( \frac{1}{8} \), which is a straightforward fraction. Recognizing this equivalence is crucial for simplifying expressions with exponents.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$\left(w^{-7}\right)^{2}\left(w^{3}\right)^{4}$$

Simplify. Write the answers with positive exponents only. $$5^{4} 5^{-7}$$

Subtract the polynomials. $$\left(9 p^{3}-4 p^{2}+2 p\right)-\left(2 p^{3}+6 p^{2}-3 p\right)$$

Simplify. Write the answers with positive exponents only. $$8^{2} 8^{-4}$$

Simplify. $$\left(\frac{2}{3}\right)^{2}\left(\frac{2}{3}\right)^{5}$$
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