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Chapter 7: Problem 28

Simplify each of the following expressions as completely as possible. Final answers should be expressed with positive exponents only. (Assume that all variables represent positive quantities.) $$\left(x^{-4}\right)^{3}$$

Short Answer

Expert verified
\( \frac{1}{x^{12}} \)

Step by step solution

01

Apply the Power Rule

Use the power rule \[ (a^m)^n = a^{mn} \] to simplify the expression \[(x^{-4})^3 \]. According to this rule, multiply the exponents together: \[(x^{-4})^3 = x^{-4 \times 3} = x^{-12} \]
02

Convert to Positive Exponent

Since the final answer should be expressed with positive exponents, rewrite \(x^{-12}\) using the property \[(a^{-m} = \frac{1}{a^m})\]. \[ x^{-12} = \frac{1}{x^{12}} \]

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Key Concepts

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

Power Rule
The Power Rule is a fundamental concept in algebra that simplifies expressions involving exponents. When you have an exponent raised to another exponent, like \((a^m)^n\), you can simplify it by multiplying the exponents together. This rule can be stated as \[ (a^m)^n = a^{mn} \]. For example, in the given exercise, we have \((x^{-4})^3\). To simplify this, use the Power Rule and multiply the exponents: \((x^{-4})^3 = x^{-4 \times 3} = x^{-12}\). This step helps in reducing complex expressions into a more manageable form.
Negative Exponents
Negative exponents can often be confusing, but they follow a simple rule that makes them easier to handle. The key property for negative exponents is \[a^{-m} = \frac{1}{a^m}\]. This means that any base raised to a negative exponent is the same as the reciprocal of that base raised to a positive exponent. In our example, after using the power rule, we get \[x^{-12}\]. To express this with positive exponents only, we rewrite it using the negative exponent rule: \[x^{-12} = \frac{1}{x^{12}}\]. This conversion helps in making the expression valid as per the requirement to use only positive exponents.
Positive Exponents
Using positive exponents is crucial in final answers for clarity and standardization. A positive exponent indicates how many times the base is multiplied by itself. For instance, \(x^3\) means \(x \times x \times x\). In simplified expressions, converting negative exponents to positive ones ensures that the final answer is in an easily interpretable form. From our solution, \((x^{-4})^3\) first simplified to \(x^{-12}\), and converting \(x^{-12}\) to a positive exponent using the previously discussed rules resulted in \(\frac{1}{x^{12}}\). By always aiming for positive exponents in our answers, we achieve a consistent format that is aligned with standard algebraic practices.

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Most popular questions from this chapter

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Do your computation using scientific notation. $$\frac{80}{0.0002}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x+4)(x-7)$$

Do your computations using scientific notation. Atomic measurements or other very small distances are often measured in units called angstroms. One angstrom, which is written \(1 \AA\), is equal to \(0.00000001 \mathrm{cm} .\) Write this number in scientific notation.

If a square has a side of \(x \mathrm{cm}\) and the side is decreased by \(6.1 \mathrm{cm},\) express the change in area in terms of \(x .\)
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