Problem 8 - Intermediate Algebra

Question

Simplify each numerical expression. \(\left(\frac{1}{2}\right)^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is 8.

Step 1: Understand the Expression

We need to simplify the expression \( \left(\frac{1}{2}\right)^{-3} \). This expression involves an exponent, which is negative.

Step 2: Apply the Negative Exponent Rule

The negative exponent rule states \( a^{-n} = \frac{1}{a^n} \). Apply this rule to \( \left(\frac{1}{2}\right)^{-3} \), which becomes \( \frac{1}{\left(\frac{1}{2}\right)^3} \).

Step 3: Simplify the Denominator

Now simplify \( \left(\frac{1}{2}\right)^3 \). This is \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).

Step 4: Simplify the Overall Expression

Put \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \) back into the expression: \( \frac{1}{\frac{1}{8}} \). To simplify this, flip the fraction: \( 8 \).

Step 5: Final Answer

After simplifying, we find that \( \left(\frac{1}{2}\right)^{-3} = 8 \).

Key Concepts

Exponent Rules

When dealing with exponents, understanding the rules is key to solving problems effectively. Exponents denote how many times a number, known as the base, is multiplied by itself. In some cases, you might come across negative exponents, like in the expression \( \left(\frac{1}{2}\right)^{-3} \). The rule for negative exponents is crucial: \( a^{-n} = \frac{1}{a^n} \). This means a negative exponent indicates the reciprocal of the base raised to the positive power.

• Example: For \( 3^{-2} \), apply the rule: \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \).
• Negative exponents can simplify seemingly complex fractions into more manageable numbers.
• Always switch the sign of the exponent and take the reciprocal of the base.

This helps in transforming expressions to a simpler format, which is a big part of math simplification.

Simplifying Expressions

Simplifying expressions is like cleaning up cluttered math problems. You take what's complicated and make it simple. In our example, we began with \( \left(\frac{1}{2}\right)^{-3} \). To simplify, first apply the negative exponent rule to turn it into \( \frac{1}{\left(\frac{1}{2}\right)^3} \).

• Next, deal with any remaining operations or calculations. Here, that means simplifying \( \left(\frac{1}{2}\right)^3 \) by calculating \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \), which equals \( \frac{1}{8} \).
• You're left with \( \frac{1}{\frac{1}{8}} \), which simplifies further. Flipping the fraction, we see that \( \frac{1}{\frac{1}{8}} = 8 \).
• Simplification often involves a series of steps like these: applying rules, making calculations, and re-arranging parts of the equation.

Through simplification, we reduce a complex problem into a neat, final result.

Fractional Exponents

Fractional exponents are another powerful tool in handling expressions involving powers. Unlike whole number exponents, fractional exponents offer greater flexibility and can express roots in a computational way.

• With a fractional exponent of the form \( a^{\frac{m}{n}} \), the base \( a \) is both raised to the power \( m \) and, subsequently, the \( n \)-th root is taken.
• This bridges multiplication and roots operations, giving you a compact way to handle them simultaneously.
• For example, \( 9^{\frac{1}{2}} \) is the same as \( \sqrt{9} \), which equals 3.
• Fractional exponents are handy in many areas, including higher-level algebra, calculus, and science.

Grasping fractional exponents is essential for a comprehensive understanding of exponent rules and helps expand your problem-solving toolkit.