Problem 17 - Algebra for College Students

Question

Evaluate each numerical expression. \((-1)^{\frac{7}{3}}\)

Step-by-Step Solution

Answer

The value of \((-1)^{\frac{7}{3}}\) is -1.

Step 1: Understand the Expression

The given expression is \((-1)^{\frac{7}{3}}\). This means -1 is raised to the power of the fractional exponent \(\frac{7}{3}\).

Step 2: Simplify the Exponent

The exponent \(\frac{7}{3}\) can be rewritten as \(2 + \frac{1}{3}\). This implies that \((-1)^{\frac{7}{3}}\) is equivalent to \((-1)^2 \cdot (-1)^{\frac{1}{3}}\).

Step 3: Calculate the Integer Power

Calculate \((-1)^2\): \((-1)\times(-1) = 1\).

Step 4: Analyze the Fractional Part

\((-1)^{\frac{1}{3}}\) represents the cube root of -1, which is also -1 since \((-1) \times (-1) \times (-1) = -1\).

Step 5: Combine the Results

Multiply the results of Step 3 and Step 4: \(1 \times (-1) = -1\).

Key Concepts

Powers of Negative Numbers

Negative numbers can behave in interesting ways when raised to different powers. A negative number raised to an integer power will result in either a positive or negative number, depending on whether the power is even or odd. Here are some helpful points to remember when dealing with powers of negative numbers:

• Negative Number to an Even Power: The product is positive because multiplying two negative numbers together gives a positive result. For instance, \[ (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16. \]
• Negative Number to an Odd Power: The product remains negative because multiplying an odd number of negative numbers together results in a negative outcome. For example, \[ (-3)^3 = (-3) \times (-3) \times (-3) = -27. \]
• Fractional Exponents: When a negative number is raised to a fractional power, the computation includes both taking a root and a power, adding complexity. This often involves deeper understanding, especially when roots (like cube roots) are included.

Simplifying Exponents

Exponents represent a number multiplied by itself a certain number of times. Simplifying exponents, especially fractional ones, is essential in many mathematical problems. Here is how you can simplify fractional exponents:

• Understanding Fractional Exponents: An exponent that is a fraction represents both a power and a root. For example, \[ a^{m/n} \] refers to taking the nth root of \(a\) and then raising it to the mth power.
• Simplify Step by Step: Break down the exponent into its components, \[ a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m. \] This will help you systematically manage complex calculations.
• Combine Like Terms: Factor out the greatest common exponent when possible to simplify your expression.

In the provided exercise, \[ (-1)^{\frac{7}{3}} = (-1)^{2} \times (-1)^{\frac{1}{3}}, \] which combines integer powers with fractional powers, simplifying it step-by-step clarifies the process.

Cube Roots

Cube roots are the special fractional exponents that represent taking the third root of a number. The cube root of a number \(a\) is written as \[ a^{\frac{1}{3}}. \]

• Understanding Cube Roots: Finding the cube root of a number means identifying a number that, when multiplied by itself twice, results in the original number. For instance, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
• Cube Roots of Negative Numbers: Interestingly, negative numbers can also possess real cube roots. This is because multiplying a negative number by itself three times still yields a negative product. For example, the cube root of -27 is -3, since \[ (-3) \times (-3) \times (-3) = -27. \]

In our example with (-1)^{\frac{1}{3}}, the result is -1, which confirms that \[ (-1) \times (-1) \times (-1) = -1. \]This shows how cube roots can be user-friendly even with negative numbers.