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Chapter 6: Problem 520

Simplify. (a) \(m^{-4}\) (b) \(\left(x^{3}\right)^{-4}\)

Short Answer

Expert verified
Part (a): \(\frac{1}{m^4}\)Part (b): \(\frac{1}{x^{12}}\)

Step by step solution

01

Understanding Negative Exponents

A negative exponent indicates that the base should be reciprocal. Essentially, for any non-zero number a and integer n, the rule is: \[a^{-n} = \frac{1}{a^n}\]
02

Simplify Part (a)

Given: \[m^{-4}\]Apply the negative exponent rule: \[m^{-4} = \frac{1}{m^4}\]
03

Simplify Part (b)

Given: \[\left(x^3\right)^{-4}\]Apply the power of a power rule first, which states \((a^m)^n = a^{m \cdot n}\), then use the negative exponent rule:\[\left(x^3\right)^{-4} = x^{3 \cdot (-4)} = x^{-12}\]Now apply the negative exponent rule: \[x^{-12} = \frac{1}{x^{12}}\]

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Key Concepts

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

Exponent Rules
When dealing with exponents, there are several fundamental rules to remember. These help us simplify expressions and solve problems efficiently. Here are the primary exponent rules you should know:
  • Product of Powers Rule: When multiplying two exponents with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\)
  • Quotient of Powers Rule: When dividing two exponents with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a Power Rule: When raising an exponent to another exponent, you multiply the exponents: \((a^m)^n = a^{m\cdot n}\)
  • Zero Exponent Rule: Any nonzero base raised to the power of zero is 1: \(a^0 = 1\)
  • Negative Exponent Rule: A negative exponent indicates taking the reciprocal of the base raised to the positive version of the exponent: \(a^{-n} = \frac{1}{a^n}\)
Understanding these rules helps in simplifying complex exponent expressions.
Reciprocal
The concept of a reciprocal is crucial when dealing with negative exponents. The term 'reciprocal' simply means to invert a number or fraction. Think of it as flipping a number upside down. For example, the reciprocal of 5 is \(\frac{1}{5}\). The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). When you have a negative exponent, you take the reciprocal of the base.
For example:
  • Given \(m^{-4}\): The negative exponent rule states \(m^{-4} = \frac{1}{m^4}\)
  • Given \((x^3)^{-4}\): First, apply the power of a power rule to get \((x^3)^{-4} = x^{-12}\), then apply the negative exponent rule to get \(\frac{1}{x^{12}}\)
This means that a negative exponent tells you to take the reciprocal of the base raised to the corresponding positive exponent.
Power of a Power Rule
The power of a power rule is used when you have an exponent raised to another exponent. This rule states that you simply multiply the exponents: \((a^m)^n = a^{m \cdot n}\). This rule helps to simplify expressions involving multiple exponents.
Let's look at an example:
  • Given \((x^3)^{-4}\): Apply the power of a power rule to get \(x^{3 \cdot (-4)} = x^{-12}\)
  • Then, use the negative exponent rule to simplify further: \(x^{-12} = \frac{1}{x^{12}}\)
By understanding and applying the power of a power rule, you can tackle expressions with nested exponents efficiently. This simplifies the process and allows you to correctly solve problems involving multiple layers of exponents.

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