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Chapter 4: Problem 65

Evaluate each expression. $$ \frac{1}{4^{-2}} $$

Short Answer

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16.

Step by step solution

01

- Understand the Negative Exponent Rule

The negative exponent rule states that for any nonzero number a and integer n, \( a^{-n} = \frac{1}{a^n} \) and \ \ \( \frac{1}{a^{-n}} = a^n \). This means we can rewrite the expression with a positive exponent.
02

- Apply the Negative Exponent Rule

Rewrite \ \ \ \( \frac{1}{4^{-2}} \) \ \ \ \( = 4^2 \). The negative exponent in the denominator moves to the numerator, becoming a positive exponent.
03

- Calculate the Exponent

Evaluate \ \ \( 4^2 \). This means multiplying 4 by itself: \ \ \( 4 \times 4 = 16 \).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number that gets multiplied, and the exponent tells how many times to multiply the base by itself. For example, in the expression \[ 4^2 \], 4 is the base and 2 is the exponent. It means 4 is multiplied by itself 2 times, or \[ 4 \times 4 = 16 \]. Exponentiation is a key concept in algebra and many other areas of math. It can simplify expressions and is essential for understanding more advanced topics. When dealing with exponents, always remember the power rules, like the product of powers (n\frac{a^m}{a^n}=a^{m-n}), which help in simplifying expressions efficiently.
Negative Exponents
Negative exponents can be confusing at first, but they follow a simple rule. The negative exponent rule states \[ a^{-n} = \frac{1}{a^n} \], where a is a nonzero number and n is a positive integer. This means a negative exponent turns the number into its reciprocal. For example, \[ 4^{-2} = \frac{1}{4^2} \], which simplifies to \[ \frac{1}{16} \]. In the given exercise \[ \frac{1}{4^{-2}} \], we apply this rule in reverse to get \[ 4^2 = 16 \]. This happens because the negative exponent in the denominator can be moved to the numerator as a positive exponent. Mastering this rule makes solving problems with negative exponents much easier.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation of algebra. Understanding how to manipulate and simplify these expressions is crucial. For example, in \[ \frac{1}{4^{-2}} \], the expression can be simplified using exponent rules. By applying the negative exponent rule, we change the expression to \[ 4^2 = 16 \]. Algebraic expressions often require several steps to simplify, involving rules like those for exponents, distributive property, and combining like terms. Being comfortable with these rules helps solve algebraic expressions efficiently and correctly.

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