Problem 41 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$32^{0.2}$$

Step-by-Step Solution

Answer

32^{0.2} \approx 2.32, depending on your calculator display.

Step 1: Understanding the Expression

The expression given is \(32^{0.2}\). This means we need to calculate the value of 32 raised to the power of 0.2. The exponent 0.2 is the same as \(\frac{1}{5}\), so we are finding the 5th root of 32.

Step 2: Using the Calculator

Turn on your calculator and access the function that allows you to calculate powers. Enter '32' and then the power operation (usually denoted by \(x^y\) or similar). Input '0.2' as the power. Press the equal or enter button to get the result.

Step 3: Interpreting the Result

The calculator should compute the value and display it on the screen. Since your display shows this value in digits, take note of all the digits shown.

Key Concepts

Fifth Root

The fifth root of a number is a special type of root. It involves finding a number which, when multiplied by itself five times, gives the original number. For example, when we talk about the fifth root of 32, we are looking for a number, that when raised to the power of 5, will result in 32.
The fifth root is represented in exponent form as the fraction 1/5 or as an exponent of 0.2. In mathematical terms, you can write this as:

• To find the fifth root of 32, we calculate: \[32^{\frac{1}{5}} = 32^{0.2}\]

Taking roots like these are fundamental in simplifying complex expressions or solving equations, where radicals are present. Understanding how to work with roots using fractional exponents can greatly simplify working with algebraic expressions.

Calculator Usage in Algebra

Using a calculator effectively is a skill that can save a lot of time, especially for complex calculations such as finding roots or powers. Calculators usually have a button or function for exponentiation, often labeled as \(x^y\). Here's how you can navigate calculator usage efficiently:

• First, make sure your calculator is on and set to the correct mode (often a "scientific" or "algebraic" mode).
• Input the base number, which in this case is 32. Press the exponent operation button next, typically labeled as \(x^y\) or with a similar symbol.
• Enter the fractional exponent, which is 0.2. This represents the fifth root in our example.
• Press the equal sign to calculate and view the result.

Every model of calculator might differ slightly, so ensure you are familiar with the operations specific to your device. Practice with a few different numbers to get the hang of it can be really beneficial.

Understanding Radical Expressions

Radical expressions, which include roots, are expressions that contain a root symbol. Understanding them is key in algebra as they appear frequently in equations and functions.
A radical expression like the fifth root of 32 builds on the concept of powers with fractional exponents. By rewriting roots as fractional exponents, such as \[\sqrt[5]{32} = 32^{0.2},\]we simplify the arithmetic processes involved since operations on exponents are straightforward with calculators.

• Radical symbols display the root and the radicand (the number inside the radical symbol). With fractional exponents, we leverage calculator functions that operate with ease on these expressions.
• Radicals allow us to express decay, growth, and many natural phenomena in a mathematical form, so mastering manipulation and simplification of these expressions is highly useful.

In summary, working with radicals often involves skills in converting between radicals and exponent forms, calculator efficiency, and interpreting the physical meaning of these mathematical expressions.