Problem 11 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(\frac{1}{2}\right)^{-3}$$

Step-by-Step Solution

Answer

8

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). Therefore, apply this rule to \( \left( \frac{1}{2} \right)^{-3} \): \[ \left( \frac{1}{2} \right)^{-3} = \frac{1}{\left( \frac{1}{2} \right)^3} \]

Step 2: Simplify the Denominator

Now, calculate \( \left( \frac{1}{2} \right)^3 \). To find this, raise \( \frac{1}{2} \) to the power of 3: \[ \left( \frac{1}{2} \right)^3 = \frac{1^3}{2^3} = \frac{1}{8} \]

Step 3: Evaluate the Expression

Having computed \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \), substitute back into the expression: \[ \frac{1}{\left( \frac{1}{2} \right)^3} = \frac{1}{\frac{1}{8}} \] The reciprocal of \( \frac{1}{8} \) is 8.

Key Concepts

Negative Exponent Rule

When faced with a number or expression raised to a negative exponent, the Negative Exponent Rule becomes a handy tool. This rule tells us that any number with a negative exponent can be converted into a fraction with a positive exponent. In simpler terms: if you see a negative power, flip it into a fraction.

For instance, if you have an expression like \((a^{-n})\), you can rewrite it as \(\frac{1}{a^n}\). This transformation makes calculations easier because it's easier to work with positive exponents. Doing this gives you a clearer picture of the expression and simplifies further operations.

Using this rule, \(\left(\frac{1}{2}\right)^{-3}\) becomes \(\frac{1}{\left(\frac{1}{2}\right)^3}\), making it simpler to compute and understand.

Positive Real Numbers

Positive real numbers are values that are greater than zero and can include both whole numbers and fractions. These are the numbers you're most familiar with, like 1, 2, 3, 1/2, 3.5, etc. In mathematical expressions, simplifying with positive real numbers ensures that our results are non-negative and tangible.

If a problem states that variables represent positive real numbers, this is crucial for ensuring solutions remain valid. It helps avoid situations that might cause confusion, like square roots of negative numbers, which are not real.

Whenever you deal with exponents and expressions in math homework or exercises, always see if the context requires positive real numbers. This condition affects how you simplify and compute numerical results. For example, understanding that \(\left(\frac{1}{2}\right)^{-3}\) is a positive real number helps guide your steps logically to the solution.

Fractional Exponents

Fractional exponents represent another approach to express roots and powers. Often, you may see square roots, cube roots, and other roots written using exponents that are fractions. This technique simplifies calculations and makes the notation compact.

A fractional exponent such as \(a^{1/n}\) equates to the nth root of \(a\). For example, \(x^{1/2}\) is the same as \(\sqrt{x}\), and \(x^{1/3}\) is equivalent to \(\sqrt[3]{x}\). When numbers or expressions are raised to fractional exponents, they unlock a pattern that is usually expressed through radical notation.

• Fractional exponents often appear when simplifying complex expressions or solving equations.
• They provide a way to handle roots using exponent rules, making calculations more straightforward.
• Recognizing fractional exponents helps identify simpler methods for solving equations that involve radicals or roots.

In the context of an expression like \(\left(\frac{1}{2}\right)^3\), although this isn’t directly a fractional exponent, understanding fractional exponents can guide comprehension of how powers and roots interplay in expressions.