Problem 3 - Algebra for College Students

Question

Evaluate each numerical expression. \(27^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The expression \(27^{\frac{1}{3}}\) evaluates to 3.

Step 1: Understand the Expression

The expression \(27^{\frac{1}{3}}\) involves finding the cube root of 27. The fraction exponent \(\frac{1}{3}\) indicates the operation of taking the cube root.

Step 2: Rewrite as a Radical

Rewrite the expression \(27^{\frac{1}{3}}\) as a radical expression: \(\sqrt[3]{27}\). This means we're looking for a number that, when multiplied by itself three times, gives 27.

Step 3: Determine the Cube Root of 27

Find the cube root of 27 by determining which number, when multiplied by itself three times (cubed), equals 27. We have: \(3 \times 3 \times 3 = 27\). Therefore, \(\sqrt[3]{27} = 3\).

Step 4: Confirm the Solution

Verify the result by cubing 3: \(3^3 = 3 \times 3 \times 3 = 27\). Since this expression equals 27, \(27^{\frac{1}{3}} = 3\) is confirmed to be correct.

Key Concepts

Fractional Exponents

Fractional exponents are an alternative way to represent roots. When you encounter an exponent expressed as a fraction, it indicates a root in the expression. Let's break down what this means. In the expression \(27^{\frac{1}{3}}\), the numerator \(1\) represents that we're dealing with the first power of the cube root, while the denominator \(3\) indicates that it is a cube root specifically.

• The top number in the fraction (numerator) is the power to which the base is raised.
• The bottom number (denominator) tells you which root to take.

So, \(a^{\frac{m}{n}}\) means you are taking the \(n\)-th root of \(a\) and then raising it to the \(m\)-th power. In our exercise, since \(m\) is \(1\), we simply take the cube root of \(27\) without any additional powers.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and others. In mathematics, converting between a fractional exponent and a radical expression is commonplace. Consider the expression \(27^{\frac{1}{3}}\) from the exercise. It can be rewritten as a radical expression: \(\sqrt[3]{27}\).

• "\(\sqrt{}\)" (the radical sign) represents the square root unless a different root is specified.
• Here, with cube roots, the small "3" written above and to the left of the radical sign indicates you are taking the cube root.

This means we're seeking a number that, when multiplied by itself three times, gives us \(27\). The operation of transforming \(27^{\frac{1}{3}}\) to \(\sqrt[3]{27}\) illustrates how fractional exponents and radical expressions are closely related.

Basic Algebra

Basic algebra often involves simplifying expressions and solving simple equations like finding cube roots efficiently. Understanding the connection between different forms of expressing numbers, such as exponents and roots, is essential. In our exercise, the goal was to evaluate \(27^{\frac{1}{3}}\). This is a straightforward example of applying basic algebra.

• "Evaluate" means to calculate the numerical value of an expression.
• Translating an expression with fractional exponents into a form where calculation is simpler, like using radicals, helps to easily find solutions.

Here, knowing the basic multiplication facts aids in recognizing that \(3 \times 3 \times 3 = 27\), simplifying the evaluation process. The entire exercise reinforces the idea of interpreting and solving expressions methodically, a fundamental skill in algebra.