Problem 7 - Elementary Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ \frac{1}{2^{-4}} $$

Step-by-Step Solution

Answer

The value is 16.

Step 1: Understand the Problem

We need to evaluate the expression \( \frac{1}{2^{-4}} \). This involves dealing with the negative exponent in the denominator.

Step 2: Apply the Negative Exponent Rule

The rule for negative exponents is \( a^{-n} = \frac{1}{a^n} \). Applying this to the denominator, we have: \[ 2^{-4} = \frac{1}{2^4} \] Thus, our expression becomes: \[ \frac{1}{\frac{1}{2^4}} \]

Step 3: Simplify the Fraction

When you have a fraction of the form \( \frac{1}{\frac{1}{a}} \), it simplifies to \( a \) because the fractions cancel each other out. Applying this to our expression gives us: \[ 2^4 \]

Step 4: Calculate the Power

Now calculate \( 2^4 \). This means multiplying 2 by itself four times: \[ 2^4 = 2 \times 2 \times 2 \times 2 = 16 \]

Step 5: Conclusion

We have simplified and calculated the expression. The value of \( \frac{1}{2^{-4}} \) is 16.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It involves reducing a fraction to its simplest form, which means that the numerator and the denominator have no common factors other than 1. In many cases, this can make complex calculations much easier to manage.
When we simplify fractions like \( \frac{1}{\frac{1}{a}} \), we leverage the idea that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \( \frac{1}{\frac{1}{a}} \) simplifies directly to \( a \). This is because when you divide by a fraction, you essentially "flip" it and multiply, removing the complexity of the expression.
To easily identify when simplification is possible, look for:

• Fractions inside fractions (complex fractions)
• Common factors in the numerator and denominator
• Cases where the numerator is 1, which can often simply flip the fraction

This clear understanding can help you tackle problems involving fractions with efficiency.

Evaluating Expressions

Evaluating expressions means finding the numerical value or solution of a given expression. This process involves performing operations according to mathematical rules and conventions, such as the order of operations.
In the exercise given, the expression \( \frac{1}{2^{-4}} \) requires evaluating by converting all elements to their simplest forms. First, recognize and apply rules like the negative exponent rule to transform the expression into one that’s easier to interpret.
Steps in evaluating expressions include:

• Identify the operations and rules applicable (e.g., exponent rules, fraction simplification).
• Perform each operation step-by-step, especially in complex expressions.
• Re-evaluate and double-check calculations for accuracy.

By understanding each operation's impact, you can confidently evaluate expressions and find the correct result.

Powers of Numbers

Powers of numbers, or exponents, are a way of representing repeated multiplication of a number by itself. For instance, \( 2^4 \) means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 \). This results in the value 16.
Understanding powers is crucial as they frequently appear in both simple and complex mathematical problems. Powers can also be used to express very large or very small numbers, often utilizing scientific notation.
Key points to know about powers:

• Positive exponents indicate how many times to multiply a number by itself.
• Negative exponents, like \( a^{-n} \), can transform a power into a fraction: \( a^{-n} = \frac{1}{a^n} \).
• Zero power, \( a^0 \), always equals 1, no matter what \( a \) is, provided \( a eq 0 \).

Recognizing these properties of powers will help simplify and solve mathematical expressions with greater ease.