Problem 107 - Elementary and Intermediate Algebra

Question

Use a calculator to evaluate each expression. Round to the nearest hundredth. See Using Your Calculator: Rational Exponents. $$ (1.045)^{2 / 5} $$

Step-by-Step Solution

Answer

The expression \((1.045)^{2/5}\) evaluates to approximately 1.02 when rounded to the nearest hundredth.

Step 1: Identify Base and Exponent

We need to evaluate the expression \((1.045)^{2/5}\). Here, the base is 1.045, and the exponent is \(\frac{2}{5}\). This means we will be taking the fifth root of 1.045 and then squaring the result.

Step 2: Use the Calculator to Find the Fifth Root

Enter 1.045 into your calculator. Most calculators have a function for finding roots, or you can use the exponentiation function with fractions. Calculate \(1.045^{1/5}\) to find the fifth root. The result should be approximately 1.0089.

Step 3: Square the Result

Now, take the result from Step 2, which is approximately 1.0089, and square it. Enter 1.0089 into the calculator and press the square function or multiply it by itself. The result should be approximately 1.018.

Step 4: Round the Answer

The result from Step 3 is approximately 1.018. To round to the nearest hundredth, look at the third decimal place (the thousandth place). Since the number is 8, which is 5 or more, you round up the second decimal place from 1.01 to 1.02.

Key Concepts

Exponentiation

Exponentiation is simply a mathematical operation involving two numbers. We call them the base and the exponent. The base is the number that is being multiplied, and the exponent indicates how many times the base is multiplied by itself. For instance, in the expression

• \(a^n\),

\(a\) is the base and \(n\) is the exponent.
In cases where the exponent is a fraction, like

• \(\frac{2}{5}\),

we are dealing with a rational exponent. A rational exponent means you are performing two operations: root calculation and exponentiation. For an expression like

• \((1.045)^{2/5}\),

it means first we take the fifth root of the base (\(1.045\)) and then square the result.
This dual operation might seem tricky, but it follows straightforward steps and can be simplified using calculators efficiently.

Root Calculation

Calculating roots is an essential part of working with rational exponents. When you see an exponent expressed as a fraction, like

• \(\frac{2}{5}\),

the denominator tells us which root to take, while the numerator tells us the power to raise the number to after finding the root.
In

• \((1.045)^{2/5}\),

we are first tasked with calculating the fifth root of 1.045. Many calculators allow you to find roots by either using a root function or the exponentiation feature with fractions. To compute \(1.045^{1/5}\), input the base (1.045) and then use your calculator's functions to find the fifth root. This should yield approximately 1.0089.
Understanding root calculation will simplify many complex operations with rational exponents.

Using a Calculator

Using a calculator effectively requires knowing the available functions and how to apply them to mathematical problems. First, identify what operations your problem requires. In dealing with rational exponents like \((1.045)^{2/5}\), you need to do both root extraction and exponentiation.
Start by entering the base number (1.045) into your calculator. Depending on your calculator, finding roots might involve pressing a key labeled something like "\(x^y\)", "root", or you may enter the root as a fractional exponent (like \(^1/5\) for the fifth root). After calculating the root, your next job is to square this result. Input the derived number—approximately 1.0089—and apply the square function or multiply it by itself.

• Ensure precision by rounding your final result correctly. In this case, the number rounds from 1.018 to 1.02.

Practice and familiarity with the tools allow you to navigate even the trickiest rational exponent problems with ease.