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Chapter 0: Problem 34

Simplify each exponential expression. $$\left(x^{-6}\right)^{4}$$

Short Answer

Expert verified
The simplified form of the expression \(\left(x^{-6}\right)^{4}\) is \(1/x^{24}\).

Step by step solution

01

Apply the law of exponents

First, the law of exponents \((a^m)^n = a^{mn}\) is applied. The power outside the bracket, which is 4, multiplies with the power inside the bracket,which is -6. So \(x^{-6}\) raised to the power of 4 will be \(x^{-6*4}\).
02

Multiply the powers

After multiplying the powers -6 and 4, the expression becomes \(x^{-24}\).
03

Simplify the expression

An exponent of -n is equivalent to 1 divided by the base raised to n. Meaning \(x^{-n} = 1/x^n\). Therefore, \(x^{-24}\) simplifies to \(1/x^{24}\).

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Key Concepts

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

Laws of Exponents
The laws of exponents are fundamental rules that help us deal with exponential expressions. These rules make operations on powers easy and systematic. Let's understand the key ones used in the exercise:

  • Power of a Power Rule: This rule states that when you raise a power to another power, you multiply the exponents. Mathematically, it's written as \((a^m)^n = a^{mn}\). In our exercise, we have \((x^{-6})^4\), which becomes \(x^{-6 \times 4} = x^{-24}\).
  • Power of a Product Rule: Though not directly used here, it's useful to know. When you raise a product to a power, distribute the exponent to each base. Expressed as \((ab)^n = a^n b^n\).
  • Power of a Quotient Rule: Similar to the product rule, when a quotient is raised to a power, apply the exponent to both the numerator andthe denominator: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).
Mastering these rules simplifies working withdifficult expressions.The laws allow you to rewrite expressions in a more workable form.
Negative Exponents
Negative exponents might seem confusing at first, but understanding them is quite simple. A negative exponent indicates reciprocal action of the base raised to the positive version of the exponent.

Consider \(x^{-n}\):
  • Here, the negative exponent \(-n\) tells us to take the reciprocal of the base (\(x\)) and apply the positive exponent. Thus, \(x^{-n}\) = \(\frac{1}{x^n}\).This idea allows us to move bases with negative exponents from the numerator to the denominator or vice versa.

    For instance,\(x^{-24}\) from our exercise becomes \(\frac{1}{x^{24}}\).
  • Understanding negative exponents allows for easy swapping of terms between parts of a fraction. This process is fundamental for simplifying complex expressions.
Remember, concepts of negative exponents transform what seems complicated into something much simpler.
Simplification of Expressions
Simplification is the process of rewriting an expression in a simpler or more easily understandable form. For exponential expressions, it often involves using laws of exponents and handling negative exponents effectively.

When simplifying, follow these necessary steps:
  • Identify and Apply Exponent Rules: Utilize rules like power of a power and negative exponent rules. They provide a straightforward path to simplify expressions.
  • Rewrite in Standard Form: Remove negative exponents by taking reciprocals. This converts expressions like \(x^{-24}\) to \(\frac{1}{x^{24}}\).benefiting clarity and further calculations.
  • Check Your Work: Cross-check each step for errors. A small misstep can change the entire expression.
Simplification not only makes an expression easier to understand but also prepares it for additional operations you might need to perform. By honing this skill, tackling even the most complicated exponential expressions becomes manageable.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Perform the indicated operations. Simplify the result, if possible. $$\frac{1}{x^{2}-2 x-8} \div\left(\frac{1}{x-4}-\frac{1}{x+2}\right)$$

$$\begin{array}{l} \text { Find the exact value of } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}} \text { without } \\ \text { the use of a calculator. } \end{array}$$

Will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200\), write a simplified algebraic expression that models the rectangle's perimeter.

Describe ways in which solving a linear inequality is similar to solving a linear equation.
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