Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$81^{0.25}$$

Step-by-Step Solution

Answer

The fourth root of 81 is 3.00000.

Step 1: Understand the Expression

The expression given is \(81^{0.25}\). The exponent \(0.25\) indicates that we are looking for the fourth root of 81 because \(0.25 \equiv \frac{1}{4}\). Essentially, we need to calculate \((81^{\frac{1}{4}})\).

Step 2: Use a Calculator

Turn on your scientific calculator and enter the base number, 81.

Step 3: Apply the Exponent

Locate the power or exponent button on your calculator, usually labeled as a `^` or `x^y` button. After entering 81, press this button.

Step 4: Enter the Exponent Value

Enter the exponent value 0.25 after pressing the power button. This tells the calculator to compute the fourth root of 81.

Step 5: Compute the Result

After entering the exponent, press the `=` or `Enter` button on the calculator to get the result. The display should show the answer.

Step 6: Write Down the Result

The calculator will display the number 3.00000, showing that the fourth root of 81 is 3 with the maximum digits available on the display.

Key Concepts

Exponents

Exponents represent repeated multiplication of a base number. In this exercise, the expression involves a fractional exponent, 0.25, which can be interpreted as the reciprocal of 4 or (\( \frac{1}{4} \)). Exponents can significantly simplify expressions, making it easier to perform calculations.

When you encounter an expression like \( x^{a/b} \), this means taking the bth root of x and then raising it to the power of a. For example, in our case, \( 81^{0.25} \) is equivalent to extracting the fourth root, since 0.25 is the same as \( \frac{1}{4} \):

• The fourth root of 81 means you find the number which, when multiplied by itself four times, gives 81.
• Raising to a fractional power can be seen as a combination of taking a root and then raising to a power.

Exponents play a crucial role in both algebra and precalculus, providing a foundation for later understanding of logarithms and exponential growth.

Roots

Roots are the inverse operation of exponentiation. They are used to "undue" a power. For example, if you have \( x^2 = 9 \), taking the square root gives \( x = 3 \).For fractional exponents involving roots, understanding their equivalence is key to simplification:

• The square root is represented as \( \sqrt{x} \) or \( x^{1/2} \).
• The cube root is \( \sqrt[3]{x} \) or \( x^{1/3} \).
• The fourth root is \( \sqrt[4]{x} \) or \( x^{1/4} \).

In our example, \( 81^{0.25} \) is taking the fourth root of 81. Calculating roots simplifies complex numbers and is an essential skill in algebra and calculus. Obtaining roots can become straightforward through practice and use of calculators, especially for larger numbers.

Scientific Calculator Usage

A scientific calculator is essential for solving problems involving exponents and roots efficiently. Here’s a quick guide to calculating roots using a scientific calculator:

First, ensure your calculator is powered on. Inputting the base number is your first step. For our example, type 81. Next, locate the correct function key on your calculator. Often, this key might be labeled as `^`, `x^y`, or similar for exponent operations.

Once you identify the button:

• Press the power button following the base number.
• Input the fractional exponent, such as 0.25 in our case, to signify the operation.
• Finish by pressing the `=` or `Enter` to execute the command.

Your display should show the result. Seeing 3.00000 reaffirms the computation that the fourth root of 81 is 3.
Remember, while scientific calculators are user-friendly, familiarizing yourself with their layout and functions remains crucial. Practicing various examples helps streamline the process, making it second nature to handle more complex math tasks.