Problem 59 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ 32^{1 / 5} $$

Step-by-Step Solution

Answer

\(32^{1/5} = 2\)

Step 1: Understand the Expression

The expression given is \(32^{1/5}\), which means we need to find the fifth root of 32.

Step 2: Apply the Concept of Roots

The operation \(32^{1/5}\) is equivalent to finding a number \(x\) such that \(x^5 = 32.\)

Step 3: Estimate the Fifth Root

Since 32 is a comparatively small number, the fifth root will also be small. Testing simple values, we find that \(2^5 = 32.\)

Step 4: Conclude the Value

Thus, the fifth root of 32 is 2, meaning \(32^{1/5} = 2.\)

Key Concepts

Evaluating Expressions

Evaluating expressions involves determining the value of an expression based on the operations it contains. Expressions involving exponents, like \(32^{1/5}\), require us to understand the relationship between numbers and their powers or roots.

• Understanding Exponents: The expression \(32^{1/5}\) involves a fractional exponent, which signifies a root.
• Breaking It Down: The number 32 is raised to the power of \(\frac{1}{5}\), indicating we seek the fifth root of 32.

By recognizing that evaluating the expression means finding a number that, when multiplied by itself five times, equals 32, we can determine the value of the expression more easily.

Fifth Root

The fifth root of a number is the value that, when multiplied by itself five times, returns the original number. The notation \(a^{1/5}\) means finding the fifth root of a, and it is mathematically represented as finding \(x\) such that \(x^5 = a\).

• Why Fifth Root? The concept of a fifth root helps identify the base number raised to the fifth power to yield a given result.
• Example Insights: In our exercise, taking the fifth root of 32, we find that 2 multiplied by itself five times gives 32, so the fifth root of 32 is 2.

The use of roots in algebra allows us to reverse operations involving exponentiation, making problems like \(32^{1/5}\) manageable.

Approximation Techniques

Approximation techniques are valuable when the exact root isn't easily calculable, or you need a decimal answer. In exercises demanding an approximation to the nearest hundredth, the root value might need to be decimalized.

• Decimal Conversion: If the root isn’t a whole number, you’d round it to the nearest hundredth for precision.
• Application in Math: Approximations are common when dealing with irrational numbers or when precision to a specific decimal place is required.

For exercises like calculating \(32^{1/5}\), however, since the answer is a whole number, approximation isn't necessary. But being acquainted with approximation practices readies students for more complex problems where exact solutions aren’t feasible.