Problem 61 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ (-8)^{4 / 3} $$

Step-by-Step Solution

Answer

The expression \((-8)^{4/3}\) equals 16.

Step 1: Understand the Expression

The expression \((-8)^{4 / 3}\) involves raising -8 to a fractional exponent. This requires understanding of how to handle negative bases and fractional exponents.

Step 2: Rewrite Using Radical Form

The expression can be rewritten using radical notation: \((-8)^{4/3} = \sqrt[3]{(-8)^4}\). This means we first take the cube root of -8, and then raise the result to the power of 4.

Step 3: Calculate the Cube Root

First calculate \(\sqrt[3]{-8}\). The cube root of -8 is -2, since \((-2) \times (-2) \times (-2) = -8\).

Step 4: Raise to the Power of 4

Next, raise the result from Step 3 to the power of 4: \((-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16\).

Step 5: Approximate the Result

Since the expression results in an integer value, the answer does not require approximation beyond the decimal point. The answer is 16.

Key Concepts

Negative Bases

When dealing with expressions like \((-8)^{4/3}\), understanding negative bases is crucial. A negative base means that the number being affected is less than zero, and the base value itself has its negative sign included in the calculation of powers.

• If the exponent is an even fraction, the final result will be positive, as multiplying an even number of negative numbers gives a positive number.
• Conversely, if the exponent is an odd fraction, the result will remain negative, as an odd number of negative numbers results in a negative product.

For our problem, \[ (-8)^{4/3} \]involves an even exponent when simplified, leading to a positive result.

Radical Notation

Radical notation gives us a way to express powers as roots. Here, the expression \((-8)^{4/3}\)can be rewritten in radical form to become \(\sqrt[3]{(-8)^4}\). This transformation helps to simplify our calculation by breaking it into more manageable parts: finding a root followed by exponentiation.

• The denominator of the fraction (3 in this example) tells us which root to take, in this case, the cube root of \(-8\).
• Meanwhile, the numerator (4) indicates the power to which the result should be raised.

This method allows a clearer approach, showing how fractional exponents can be systematically handled.

Cube Root

The cube root process involves finding a number that, when multiplied by itself three times, equals the original number. This is particularly useful when evaluating expressions with fractional exponents involving cubes, like \(\sqrt[3]{-8}\).

• For \(-8\), we find that \(-2\) is the cube root since \((-2) \times (-2) \times (-2) = -8\).
• Finding cube roots of other negative numbers follows the same principle: identify the number which repeats itself threefold to match the original value.

Recognizing these relationships is critical when rewriting fractional exponents as radicals.

Integer Values

Working with integer values is another aspect of dealing with expressions like \((-8)^{4/3}\). In the solution, once the cube root is found, the next step is raising it to an integer power.

• An intermediate result from the cube root, \(-2\), is raised to the fourth power: \((-2)^4\).
• The operation \((-2) \times (-2) \times (-2) \times (-2)\) results in an integer value, which in this case, is \(16\).

When all calculations produce whole numbers, it implies the entire expression resolves to an integer. In our exercise, the final result does not require any decimal places.