Problem 4 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ (-32)^{\frac{1}{5}} $$

Step-by-Step Solution

Answer

\((-2)\) satisfies \((-32)^{\frac{1}{5}}\).

Step 1: Understand the Expression

The expression \((-32)^{\frac{1}{5}}\) involves finding the fifth root of \(-32\). This means we need to find a number that, when raised to the power of \(5\), equals \(-32\).

Step 2: Evaluate the Fifth Root Conceptually

Since the expression involves an odd root, the result can be negative. We need to determine what number multiplied by itself four more times equals \(-32\).

Step 3: Identify the Number

Start by considering smaller numbers: \((-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\). Thus, \((-2)\) is the number that satisfies the root condition.

Step 4: Verify Your Result

Check your result by confirming that \((-2)^5 = -32\), ensuring the calculation correctly evaluates to the given expression.

Key Concepts

Understanding Numerical Expressions

A numerical expression is a mathematical phrase that includes numbers, operations, and sometimes variables. It does not have an equality or inequality sign.
For example, in the given expression \((-32)^{\frac{1}{5}}\), you have a base of \(-32\) and an exponent of \(\frac{1}{5}\). This expression requires evaluating the fifth root of \(-32\).
In any numerical expression involving exponents, numbers are raised to a power. Here, the number \(-32\) is raised to the power of \(\frac{1}{5}\) indicating the fifth root. We express such roots through fractional exponents where the denominator represents the root involved. Particularly, \(\frac{1}{5}\) denotes the fifth root.

Negative Numbers and Exponentiation

When dealing with exponentiation of negative numbers, it's crucial to consider whether the exponent is an integer or a fractional power.

• If the exponent is an integer, multiplying the number by itself involves repeating the process for as many times as the exponent indicates. Powers with even integers result in positive values because multiplying a negative number by itself an even number of times yields a positive number.
• If the exponent is a fractional power, it turns into a root operation. This changes the approach because fractions represent roots, like in our expression \((-32)^{\frac{1}{5}}\).

It's important to note that fractional exponents allow us to compute roots, and they can cope with negative bases provided the root's order is odd. This concept is essential in understanding how the fifth root can exist for a negative base number.

Odd Roots and Negative Bases

When evaluating odd roots of negative numbers, keep in mind that the result will also be negative. Let's illustrate this using our example \((-32)^{\frac{1}{5}}\).
The fifth root in this case asks "What number multiplied by itself five times equals -32?". By trying \(-2\), you can see:

• \((-2) \times (-2) = 4\)
• \(4 \times (-2) = -8\)
• \(-8 \times (-2) = 16\)
• \(16 \times (-2) = -32\)

Therefore, \(-2\) is indeed the fifth root of \(-32\). The key takeaway is that odd roots of negatives remain negatives, allowing us to compute values for expressions with odd root orders.