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Chapter 4: Problem 55

\(\frac{6^{-3}}{6^{2}}\)

Short Answer

Expert verified
\frac{1}{6^5}

Step by step solution

01

Understand the Problem

The task is to simplify the expression \(\frac{6^{-3}}{6^{2}}\). The base of both the numerator and the denominator is the same, which is 6.
02

Apply the Properties of Exponents

Recall the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\). This property can be used to combine the exponents.
03

Simplify the Exponents

Subtract the exponent in the denominator from the exponent in the numerator: \(6^{-3-2} = 6^{-5}\).
04

Simplify the Expression with Negative Exponents

Recall that \(a^{-n} = \frac{1}{a^n}\). So, \(6^{-5} = \frac{1}{6^5}\).

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky, but they follow straightforward rules. When you see a negative exponent like in the original problem \[\frac{6^{-3}}{6^{2}}\], it simply represents a fraction. The rule is: \(a^{-n} = \frac{1}{a^n}\). So, for example, \(6^{-3}\) means \(\frac{1}{6^3}\).

This concept allows us to rewrite expressions with negative exponents as fractions, making them easier to handle and simplifying many algebraic equations. In the final step of our solution: \(6^{-5} = \frac{1}{6^5}\).
Properties of Exponents
Understanding the properties of exponents is key to mastering algebra. One critical property is that when we divide like bases, we subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

In our exercise, the base is 6 for both terms: \[\frac{6^{-3}}{6^{2}}\].
Using the property, we subtract the exponent in the denominator from the exponent in the numerator: \(6^{-3-2} = 6^{-5}\).

These properties help in simplifying expressions, solving equations, and understanding the behaviors and relationships between exponential terms in mathematics.
Basic Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It represents numbers and operations in general forms, allowing us to solve equations and understand relationships.

In the problem \[\frac{6^{-3}}{6^2}\], we use basic algebraic concepts like exponents and fractions.
  • Exponents represent repeated multiplication: \(6^2 = 6 \times 6\).
  • The rules of division for exponents help simplify complex expressions.

The goal is to make the expression simpler, making it easier to understand and solve.

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Most popular questions from this chapter

In 2016 , the state of Minnesota had about \(7.33 \times 10^{4}\) farms with an average of \(3.53 \times 10^{2}\) acres per farm. What was the total number of acres devoted to farmland in Minnesota that year? (Data from U.S. Department of Agriculture.)

Find each product. $$ (4 x+3)(2 y-1) $$

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During the \(2016-2017\) season, Broadway shows grossed a total of \(\$ 1.45 \times 10^{9}\). Total attendance for the season was \(1.33 \times 10^{7}\). What was the average ticket price (to the nearest cent) for a Broadway show? (Data from The Broadway League.)

Find each product. $$ -3 r(r-1)\left(r^{2}+r+1\right) $$
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