Problem 116 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ \frac{3}{2^{-3}} $$

Step-by-Step Solution

Answer

Question: Evaluate the numerical expression \(\frac{3}{2^{-3}}\). Answer: 24

Step 1: Recall the properties of exponents

When we have an exponent in the denominator like \(2^{-3}\), we can rewrite it as a positive exponent in the numerator. In general, \(\frac{1}{a^{-n}} = a^n\).

Step 2: Rewrite the denominator

Using the property from step 1, rewrite the exponent in the denominator: \(\frac{3}{2^{-3}} = \frac{3}{\frac{1}{2^3}}\)

Step 3: Simplify the expression

Now we multiply the numerator and denominator by \(2^3\) to eliminate the fraction in the denominator: \(\frac{3}{\frac{1}{2^3}} = \frac{3 \cdot 2^3}{1} = \frac{3 \cdot 8}{1}\)

Step 4: Calculate the result

Finally, we can multiply the numerator terms to obtain the result: \(\frac{3 \cdot 8}{1} = \frac{24}{1} = 24\) The numerical expression evaluates to 24.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with numerical expressions involving powers. An exponent indicates how many times a number, called the base, is multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning \(2\) is multiplied by itself \(3\) times: \(2 \times 2 \times 2\).

Several key properties can simplify expressions with exponents:

• The product of powers property dictates that when multiplying two powers with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• The power of a power property tells us to multiply exponents when a power is raised to another power: \((a^m)^n = a^{m \times n}\).
• The power of a product property states that when a product is raised to an exponent, we raise each factor to the exponent: \((ab)^n = a^n b^n\).

The comprehension of these properties allows for more efficient simplification of complex expressions.

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra, which involves reducing an expression to its simplest form. This often requires combining like terms, reducing fractions, and applying mathematical properties, including those of exponents.

To simplify effectively, follow these common strategies:

• Combine like terms, which are terms with the same variables and exponents.
• Use mathematical properties, such as distributive, associative, and commutative properties, to rearrange and combine elements in an expression.
• Reduce fractions by dividing the numerator and the denominator by their greatest common factor.

Simplification not only makes expressions easier to understand but also prepares them for further algebraic operations and solving.

Negative Exponents

Negative exponents can be perplexing at first glance, but they adhere to a simple rule that makes working with them straightforward. The expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\), meaning a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

Here are a few examples:

• \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
• \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

Remember that when you have a fraction with a negative exponent in the denominator, you can move it to the numerator and make the exponent positive, which is what occurs when simplifying the original exercise problem.

Fractions in Algebra

Dealing with fractions in algebra involves applying the same rules used with whole numbers but requires additional steps to ensure correct simplification. When evaluating expressions with fractions, keep in mind the fundamental principle that dividing by a fraction is the same as multiplying by its reciprocal.

Consider these tips when working with fractions:

• To combine fractions with different denominators, find a common denominator.
• Multiply both the numerator and the denominator by the same number to eliminate complex fractions, as seen in the exercise solution.
• When simplifying expressions with variables in fractions, cancel out common factors in the numerator and denominator to reduce the expression.

Having a systematic approach to fractions can greatly simplify the process of manipulating and evaluating algebraic expressions.