Problem 29 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \frac{1}{2^{-4}} $$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Simplify the Denominator

The expression given is \( \frac{1}{2^{-4}} \). Start by simplifying the denominator. The negative exponent indicates that the expression can be rewritten as the reciprocal of the base raised to the positive exponent. Thus, \( 2^{-4} = \frac{1}{2^{4}} \).

Step 2: Invert the Denominator

Now that we have \( 2^{-4} = \frac{1}{2^4} \), the expression becomes \( \frac{1}{\frac{1}{2^4}} \). When dividing by a fraction, we multiply by its reciprocal. Therefore, \( \frac{1}{\frac{1}{2^4}} = 2^4 \).

Step 3: Calculate \( 2^4 \)

Next, calculate \( 2^4 \). This means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \).

Step 4: Verify with a Calculator

Finally, use a calculator to ensure that \( 2^4 = 16 \) for accurate verification of the calculation.

Key Concepts

Reciprocal

Understanding reciprocals is crucial when working with negative exponents. A reciprocal simply means "flipping" the numerator and the denominator of a fraction. For example, if you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

In the context of negative exponents, a negative exponent indicates that you should take the reciprocal of the base and then raise it to the absolute value of the exponent. So, when we have \( 2^{-4} \), it means we take the reciprocal of \( 2 \) to get \( \frac{1}{2} \), and then raise it to the power of 4. This makes \( 2^{-4} = \left(\frac{1}{2}\right)^4 \).

It's important to learn about reciprocals because they help you simplify expressions with negative exponents and make solving such expressions easier.

Exponentiation

Exponentiation is a mathematical operation where a number, known as the "base," is multiplied by itself a certain number of times. This number of times is called the "exponent." For example, in the expression \( 2^4 \), 2 is the base and 4 is the exponent.

Calculating \( 2^4 \) means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \). Each time you multiply by the base, you are said to be "raising it to a power," and the expression tells you how many times this should happen.

• If the exponent is a positive integer, like 4, you multiply the base by itself that many times.
• If the exponent is negative, like -4, you will need to first find the reciprocal of the base and then raise it to the positive power value.
  

Understanding exponentiation helps simplify complex expressions and is a vital part of algebra and higher math.

Simplifying Expressions

Simplifying expressions involves rewriting them in the most concise and easily understandable form. This process often includes combining like terms and applying rules of arithmetic to make an expression as simple as possible.

When simplifying expressions with negative exponents, the rules of reciprocals and exponentiation come into play.

Steps for Simplifying:

• Identify negative exponents and apply the reciprocal.
• Convert these expressions to positive exponents by rewriting using reciprocals.
• Compute any powers present using exponentiation.
• Simplify any resulting expressions by performing basic arithmetic operations.

In our example given, \( \frac{1}{2^{-4}} \), simplifying it involves recognizing the negative exponent. We first rewrite it using the reciprocal, giving us \( 2^4 \), and finally computing this to get 16. Simplifying expressions ensures clearer results and easier calculations.