Problem 6 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$27^{4 / 3}$$

Step-by-Step Solution

Answer

\(27^{4/3} = 81\).

Step 1: Understand the Expression

The expression \(27^{4/3}\) involves dealing with an exponent that is a fraction. Fractional exponents imply both a power and a root operation.

Step 2: Rewrite Using Root and Power

Recognize \(27^{4/3}\) as \((27^{1/3})^4\). This means finding the cube root of 27 first, and then raising the result to the power of 4.

Step 3: Find the Cube Root

The cube root of 27 is 3, since \(3^3 = 27\). Therefore, \(27^{1/3} = 3\).

Step 4: Raise to the Power 4

Now we take the result from Step 3 and raise it to the power of 4. Compute \(3^4\).

Step 5: Calculate the Final Value

Calculate \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Key Concepts

Power and Root Operations

When we encounter an expression with a fractional exponent, like \(27^{4/3}\), it's important to understand that fractional exponents involve both power and root operations. This is because the fraction represents two mathematical operations: the numerator is a power, while the denominator is a root.
For instance, in the expression \(a^{m/n}\), \(n\) signifies the root we need to take, whereas \(m\) indicates the power function to apply afterward.

• The denominator of the fraction, \(n\), tells us to calculate the nth root of the base (27), which means we find a number that, when multiplied by itself \(n\) times, equals the base.
• The numerator, \(m\), indicates that once we have the nth root, we must then raise it to the power of \(m\).

Understanding this dual operation is key to simplifying the evaluation of expressions like \(27^{4/3}\) into manageable steps.

Cube Root Calculation

Evaluating cube roots is a specific application of root operations. In our example, we have \(27^{1/3}\), which requires finding the cube root of 27.
Calculating cube roots involves determining a number that equals the original number when it is cubed—multiplied by itself three times.

• Start by considering smaller numbers that might cube to reach your original number; for 27, we recognize that \(3^3 = 27\).
• Thus, the cube root of 27 is 3, expressed as \(27^{1/3} = 3\).

These basic computations help simplify the original expression, making subsequent operations more manageable.

Exponentiation Steps

Once the root has been determined, the next step involves exponentiation, which is the process of raising a number to a power.
From our previous calculation, we have the cube root of 27 as 3. Now the task becomes raising this result, 3, to the power of 4, as indicated by \(3^4\).

• Perform sequential multiplication: Compute \(3 \times 3 = 9\), then \(9 \times 3 = 27\), and finally \(27 \times 3 = 81\).
• Thus, we've computed \(3^4 = 81\).

By following these exponentiation steps, you efficiently compute the value of such expressions, transforming complex operations into simpler tasks.