Problem 49 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. Then evaluate the expression when the result is an integer. $$ (-1)^{4 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to 1 in radical notation and as an integer.

Step 1: Understand the Expression

The given expression is \[(-1)^{4/3}\]which means raising \(-1\) to the power of \(\frac{4}{3}\). This can be split into two parts: raising to the power of 4, and taking the cube root.

Step 2: Convert to Radical Notation

We can express the fractional exponent in radical notation. The general form is \(a^{m/n} = \sqrt[n]{a^m}\). Thus, \[(-1)^{4/3} = \sqrt[3]{(-1)^4}.\]

Step 3: Simplify (-1)^4

Calculate \((-1)^4\): \[(-1)^4 = (-1) \times (-1) \times (-1) \times (-1).\]Since the number of negative signs is even, the result is positive.Thus, \[(-1)^4 = 1.\]

Step 4: Evaluate the Cube Root

Substitute back into the expression: \[\sqrt[3]{1}.\]The cube root of 1 is 1, because \[1^3 = 1.\]

Step 5: Conclude the Result

After simplifying, the result is an integer. The expression \[(-1)^{4/3}\] evaluates to 1 in radical form as well as directly evaluated.

Key Concepts

Fractional Exponents

Fractional exponents might seem tricky at first, but they are actually a powerful way to express roots and powers together. When you see an exponent as a fraction, like \( \frac{4}{3} \), it's telling you two things:

• The numerator (4 here) is the power you raise the base to.
• The denominator (3 here) is the root you take.

So, for the expression \((-1)^{4/3}\), you raise \(-1\) to the fourth power first, and then take the cube root of the result. This notation is extremely useful in mathematics because it gives you a compact way to show combined operations.
Essentially, \(a^{m/n}\) can be written as \(\sqrt[n]{a^m}\), which combines both a power and a root. This concept opens many doors for simplifying expressions and solving equations because you can approach the problem in parts.

Cube Root

The cube root is an interesting operation that involves finding a number which, when multiplied by itself three times (exumber) makes the original number. For example, the cube root of 1 is 1, because multiplying 1 by itself (1 x 1 x 1) equals 1.
When we encounter an expression like \(\sqrt[3]{1}\), it means we're looking for a number that, when raised to the power of 3, gives 1. Since \(1^3 = 1\), the cube root of 1 is unmistakably 1.
A cube root can simplify expressions and help us solve equations, especially those involving fractional exponents. While square roots deal with pairs, cube roots work with triplets, revealing their own unique properties and solutions. Understanding cube roots can enhance your problem-solving skills significantly, showing how different powers and roots can relate.

Integer Evaluation

Integer evaluation refers to simplifying an expression to a neat whole number, free of fractions or decimals. It's the final step of many math problems, where you seek to express your answer in simplest form.
In the problem \((-1)^{4/3}\), after rewriting the expression using radical notation and simplifying it, we aim to find a whole number. Upon calculating the power and root, what we get is 1. This transformation into an integer confirms that our approach and computation are correct.

• First, ensure all calculations are correct and all steps are followed accurately.
• Simplify gradually, checking each step until a whole number emerges.

The beauty of integer evaluation is in its simplicity and clarity. It assures that the expression is fully reduced and makes sense operationally in mathematical terms. Hence, mastering this skill is crucial for effective problem solving.