Problem 29 - Algebra for College Students

Question

Evaluate each numerical expression. \(125^{\frac{4}{3}}\)

Step-by-Step Solution

Answer

The value of \(125^{\frac{4}{3}}\) is 625.

Step 1: Understand the Exponential Expression

The expression we need to evaluate is \(125^{\frac{4}{3}}\). This expression is in the form of \(a^{\frac{m}{n}}\), which represents a power that combines both an exponent and a root. In general, \(a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}\).

Step 2: Break Down the Fractional Exponent

The fraction \(\frac{4}{3}\) indicates a power of 4 and a root of 3. Therefore, \(125^{\frac{4}{3}}\) can be interpreted as taking the cube root of \(125\) and then raising the result to the power of 4, or alternatively, raising \(125\) to the power of 4 and then taking the cube root.

Step 3: Find the Cube Root of 125

First, find \(125^{\frac{1}{3}}\), which is the cube root of 125. Since \(5^3 = 125\), the cube root of 125 is 5. Therefore, \(125^{\frac{1}{3}} = 5\).

Step 4: Raise the Result to the Power of 4

We now take the result from Step 3, which is 5, and raise it to the power of 4. Calculate \(5^4 = 5 \times 5 \times 5 \times 5 = 625\).

Step 5: Conclusion: Evaluate the Expression

Multiplying out the powers, we conclude that \(125^{\frac{4}{3}} = 625\), because the cube root of 125 is 5, and 5 raised to the fourth power is 625.

Key Concepts

Cube Root

A cube root of a number is a value that, when multiplied by itself twice more, gives the original number. For example, the cube root of 125 is 5, because when you multiply 5 by itself twice (i.e., 5 x 5 x 5), you get 125. Understanding cube roots can help in simplifying and solving equations involving fractional exponents. If an exponent appears as a fraction like \(\frac{1}{3}\), it implies taking the cube root of a number.

• The cube root symbol is \(\sqrt[3]{x}\).
• Recognize that the cube root is the inverse operation of cubing a number.

Knowing cube roots is crucial in solving various algebraic problems, including those involving exponential expressions.

Exponential Expressions

Exponential expressions involve numbers raised to a power or exponent. In the expression \(a^m\), \(a\) is the base, and \(m\) is the exponent. Exponents signify how many times the base number is multiplied by itself. A fractional exponent like \(\frac{m}{n}\) can be interpreted in two equivalent ways:

• One way is to first find the root of the base using \(n\), and then raise that result to the power of \(m\).
• Alternatively, raise the base to the power of \(m\) first, then take the root using \(n\).

For example, the expression \(125^{\frac{4}{3}}\) can be calculated by finding the cube root of 125 first and then raising the result to the power of 4. Exponential expressions with fractional exponents provide a useful method for working with non-integer powers and roots.

Evaluating Numerical Expressions

Evaluating numerical expressions requires understanding and applying the correct order of operations and arithmetic processes. When dealing with expressions like \(125^{\frac{4}{3}}\), you need to interpret the fractional exponent correctly and follow the steps to break it down well. Here's the basic approach:

• Identify and isolate any exponents.
• If fractional, decide the order of operations - root first or exponent first.
• Solve the root or power calculations step by step.

In practice, evaluators would first calculate the cube root of 125, finding it to be 5, and then raise 5 to the power of 4 to get 625. By following the steps precisely and methodically, one can evaluate complex numerical expressions accurately.