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Chapter 4: Problem 82

Apply the quotient rule for exponents, if possible, and write each result using only positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{8}{8^{-1}} $$

Short Answer

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Step by step solution

01

- Rewrite the expression

Start by writing the given expression as it is: \[ \frac{8}{8^{-1}} \]
02

- Apply the quotient rule for exponents

The quotient rule for exponents states \[ \frac{a^m}{a^n} = a^{m-n} \]. Applying this rule to the expression: \[ \frac{8^1}{8^{-1}} = 8^{1-(-1)} = 8^{1+1} \]
03

- Simplify the exponent

Simplify the exponent: \[ 8^{1+1} = 8^2 \]
04

- Final result

The final result with only positive exponents is: \[ 8^2 = 64 \]

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, the expression \(8^2\) means that the number 8 is multiplied by itself two times: \( 8 \times 8 \).
Exponents follow certain rules which make calculations easier. Understanding these rules is essential for simplifying more complex expressions.
Always remember that any number raised to the power of 0 is 1: \( a^0 = 1 \) if \(a eq 0\) .
Quotient Rule
The quotient rule for exponents deals with the division of two exponential expressions with the same base.
According to the quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \). In this formula, \( a \) is the base, and \( m \) and \( n \) are the exponents.
This rule simplifies the division by subtracting the exponent in the denominator from the exponent in the numerator.
Let's apply it to the initial problem: \( \frac{8}{8^{-1}} \). Here, we rewrite 8 as \( 8^1 \) to match the format: \( \frac{8^1}{8^{-1}} \). Using the quotient rule, we get: \( 8^{1-(-1)} = 8^{1+1} \).
Therefore, \( 8^{2} \).
Simplification
Simplification involves reducing an expression to its most basic form.
In our example, we already simplified the exponents from the quotient rule to get: \( 8^{2} \).
Lastly, we calculate \( 8^{2} \) to find the simplified result: \( 8 \times 8 = 64 \).
Simplification helps to avoid dealing with unnecessarily complicated expressions and makes equations easier to understand and solve.
Positive Exponents
Positive exponents indicate how many times a base number appears in a multiplication.
It's essential to write final answers using positive exponents to adhere to standard mathematical practices and improve clarity.
Notice that in the problem \( \frac{8}{8^{-1}} \), we ended up with \( 8^{2} \) after applying the quotient rule.
This is already a positive exponent, signaling a valid and suitably simplified expression.
Converting negative to positive exponents often makes the calculations easier and more intuitive.

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