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Chapter 8: Problem 84

Simplify: \(\frac{(2 x)^{5}}{x^{3}} .\) (Section 5.7, Example 5)

Short Answer

Expert verified
The simplified form of given exercise is \(32x^2\).

Step by step solution

01

Apply the power of a power rule

We begin by applying the power of a power rule to \((2x)^5\). This gives us \(2^5 \times x^5\), which simplifies to \(32x^5\). So, we now have \(\frac{32x^5}{x^{3}}\).
02

Apply the rule of exponents for division

We continue by applying the rule of exponents for division \(\frac{a^m}{a^n} = a^{m-n}\). With these values \(m=5\) & \(n=3\) so we have \(a^{m-n}\) means \(x^{5-3}\). This simplifies to \(x^2\). So, we are left with \(32x^2\).

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Key Concepts

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

Understanding the Power of a Power Rule
The power of a power rule is a fundamental concept in exponents. It is used to simplify expressions where an exponent is raised to another exponent. For example, if you have
  • \((a^m)^n\).
You multiply the exponents together to get:
  • \(a^{m \times n}\).
When we applied this rule to
  • \((2x)^5\),
we treated the term as
  • \(2^5 \times x^5\),
giving us
  • \(32x^5\).
This step sets the stage for simplifying the entire expression. By handling each part separately, we make the expression easier to work with.
Applying the Rule of Exponents for Division
The rule of exponents for division helps simplify expressions when dividing like bases with exponents. This rule states:
  • \(\frac{a^m}{a^n} = a^{m-n}\).
It means you subtract the exponent in the denominator from the exponent in the numerator. In our exercise:
  • The expression was \(\frac{32x^5}{x^{3}}\).
Applying the rule, we focused on the \(x\) terms:
  • \(x^{5-3}\),
which simplifies to
  • \(x^2\).
This simplification allows us to manage complex expressions systematically and efficiently.
Efficient Simplification of Expressions
Simplifying expressions becomes manageable when we use the rules of exponents effectively. It's like unraveling a puzzle:
  • Step 1: Break down each part using the power of a power rule.
  • Step 2: Apply the rule of exponents for division, simplifying exponent terms by subtraction.
In our initial example, we ended with
  • \(32x^2\).
This shows the power of organized simplification. Each rule has a specific function, guiding us to a cleaner, final expression. Practice these steps to improve your skills in handling exponents effectively.

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Most popular questions from this chapter

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$\left(\frac{1}{9}\right)^{-\frac{1}{2}}$$

What is the meaning of \(a^{-\frac{m}{n}} ?\) Give an example.

Rationalize the denominator: \(\frac{1}{\sqrt[3]{2}}\)

Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I rationalized a numerical denominator and the simplified denominator still contained an irrational number.

In Exercises \(94-97,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{4+8 \sqrt{3}}{4}=1+8 \sqrt{3}$$
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