Problem 24 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(\frac{1}{27}\)

Step 1: Evaluate the Inner Exponent

Start by evaluating the innermost exponent. Here, the base is 3 and the exponent is -1, which can be written as the reciprocal: \[ 3^{-1} = \frac{1}{3} \]

Step 2: Apply the Outer Exponent

Now, apply the outer exponent to the result from Step 1. Since we have \(\left(\frac{1}{3}\right)^{3}\), this means multiplying the fraction by itself three times:\[ \left(\frac{1}{3}\right)^{3} = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \]

Step 3: Calculate the Result

Multiply the fractions: First, calculate the numerator: \(1 \times 1 \times 1 = 1\).Next, calculate the denominator: \(3 \times 3 \times 3 = 27\).So, the expression simplifies to:\[ \frac{1}{27} \]

Key Concepts

Negative Exponents

Negative exponents might seem a bit tricky at first, but with a simple rule, they become easy to handle. When you see a negative exponent, it means you need to take the reciprocal of the base and make the exponent positive. For example, if you have an expression like \( 3^{-1} \), this doesn't mean you'd have a negative number but instead the reciprocal, \( \frac{1}{3} \). This rule holds true for any non-zero base \( a \), where \( a^{-n} = \frac{1}{a^n} \). By understanding this, you'll find that evaluating expressions with negative exponents becomes straightforward.
It's essential to get comfortable with this concept, as it is quite common in algebra and calculus.

Fraction Multiplication

Multiplying fractions is simpler than you might think. When you multiply fractions, you don't cross-multiply. Instead, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together directly. For example, if you have the expression \( \left(\frac{1}{3}\right)^3 \), this means you multiply \( \frac{1}{3} \) by itself three times: \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \).

• Multiply the numerators: \( 1 \times 1 \times 1 = 1 \)
• Multiply the denominators: \( 3 \times 3 \times 3 = 27 \)

The result is \( \frac{1}{27} \). So, multiplying fractions follows a straightforward path where you handle numerators and denominators separately. Remembering this process will make fraction multiplication easy to tackle.

Numerical Expressions

A numerical expression involves numbers and operations without an equal sign. Simplifying numerical expressions means you perform these operations to reach a single number. For instance, in the expression \( \left(3^{-1}\right)^3 \), we start by simplifying the exponential parts first. This entails dealing with negative exponents and applying them step by step, as shown in the detailed solution.
Performing operations systematically ensures accuracy. When simplifying, always:

• Resolve exponents first, including handling negative exponents by taking reciprocals
• Follow by completing any multiplication or division as seen in the expression

By breaking down the process, complex-looking numerical expressions become much easier to handle in bite-sized steps.