Problem 34 - Algebra for College Students

Question

Simplify each numerical expression. \(\frac{2^{-2}}{2^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{32}\).

Step 1: Understand Negative Exponents

Negative exponents indicate reciprocal. Hence, any non-zero number raised to a negative exponent can be rewritten as a reciprocal of the same number raised to the positive exponent. Therefore, \(2^{-2}\) can be rewritten as \(\frac{1}{2^2}\).

Step 2: Calculate Positive Exponents

Now, simplify the expression \(\frac{1}{2^2}\). Since \(2^2 = 4\), we have \(\frac{1}{4}\).

Step 3: Combine the Fractional Forms

Now we will substitute \(\frac{1}{4}\) for \(2^{-2}\) in the given expression to get \(\frac{\frac{1}{4}}{2^3}\).

Step 4: Simplify the Expression

Further simplify the expression \(\frac{\frac{1}{4}}{2^3}\), where \(2^3 = 8\). So it becomes \(\frac{1}{4} \times \frac{1}{8} = \frac{1}{32}\).

Key Concepts

Negative Exponents

Negative exponents can often seem tricky, but they are pretty straightforward once you get the hang of it. The key to understanding negative exponents is the concept of the reciprocal. When a number or variable is raised to a negative exponent, it implies division rather than multiplication. To simplify, just take the reciprocal of the base and switch the negative exponent to a positive one.

For example, consider the expression:

• \(2^{-2}\)

This can be rewritten as:

• \(\frac{1}{2^2}\)

Negative exponents essentially "flip" the fraction, moving the base to the opposite part of the fraction. Always remember: A negative exponent in the numerator switches places to become a positive exponent in the denominator, and vice versa.

Positive Exponents

Positive exponents signify straightforward multiplication. When a number is raised to a positive exponent, you multiply the number by itself as many times as the exponent indicates.

For instance, an expression with a positive exponent like:

• \(2^3\)

means you multiply 2 by itself three times:

• \(2 \times 2 \times 2\)

which equates to 8. Thus, positive exponents simplify conceptual multiplication. For the calculation of \(2^2\), you perform:

• \(2 \times 2 = 4\)

This concept allows us to assess values simply and directly.

Fractional Forms

Fractional forms are an elegant way to express numbers, especially when dealing with exponents. A number with a negative exponent is transformed into its positive counterpart by dividing rather than multiplying. This is simplified through fractions.

For example, let's simplify the expression:

• \(\frac{1}{2^2}\)

This gives us a fractional form of the negative exponent \(2^{-2}\).

Combining fractional forms is a big part of simplification. Let's examine the expression:

• \(\frac{\frac{1}{4}}{2^3}\)

Here, the fractional form \(\frac{1}{4}\) is divided by \(2^3\), simplifying to a multiplication of fractions: \(\frac{1}{4} \times \frac{1}{8}\), equaling \(\frac{1}{32}\). This approach helps to solve complex expressions through simple multiplication of fractions.

Exponent Rules

Exponent rules are fundamental to simplifying expressions. Understanding these laws allows for efficient simplification of expressions involving both negative and positive exponents.

Here are some key exponent rules to note:

• Product of Powers Rule: When multiplying like bases, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents. For example, \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When raising an exponent to another exponent, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).
• Negative Exponents: Already mentioned, but it’s the rule that \(a^{-n} = \frac{1}{a^n}\).

By applying these rules, you can simplify even the most complicated of numerical expressions, effortlessly managing negative, positive, and fractional exponents.