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Chapter 7: Problem 123

Use a calculator to write a four-decimal-place approximation of each number. $$ 8^{1 / 4} $$

Short Answer

Expert verified
The four-decimal-place approximation of \( 8^{1/4} \) is 1.6820.

Step by step solution

01

Understand the Expression

First, recognize the expression given is to find the fourth root of 8, which can be written as \( 8^{1/4} \). This means we want to find a number which, when raised to the power of 4, gives 8.
02

Enter the Expression in Calculator

Using a calculator, input \( 8^{(1/4)} \). On most scientific calculators, you may find a button for power, which looks like \( y^x \) or similar. Enter 8, press the power button, and then enter 0.25 (since \( 1/4 = 0.25 \)).
03

Obtain the Result

After entering \( 8^{0.25} \) into the calculator, press the equals button to get the result. The calculator should display the approximation of the fourth root of 8.
04

Round to Four Decimal Places

Check the calculator result and round it to four decimal places. The value should be approximately \( 1.682\). Since we round to four decimal places, we ensure that only four digits appear after the decimal point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourth Root Calculation
When you hear the term 'fourth root', think about reversing a power operation. If you have a number raised to the fourth power, the fourth root of that number will undo this action. For instance, if you have 8 and want to find its fourth root, you ask: "What number multiplied by itself four times equals 8?"

To find the fourth root of a number like 8, we use the expression \( 8^{1/4} \). This expression means you're looking for a number that gives you 8 when raised to the fourth power. It's like finding the square root, but instead of taking the root twice, you're doing it four times.
  • Use a scientific calculator that supports fractional powers to solve this easily.
  • Input \( 8^{0.25} \), which is the decimal equivalent of \( 1/4 \).
  • Press the power button and then equals or enter, and voila, you have your fourth root!
Decimal Place Rounding
Rounding is all about making numbers simpler while keeping them close to their original value. Sometimes calculations give results with many decimal places, which can be confusing or impractical.

When asked to round to four decimal places, focus on the first four numbers to the right of the decimal point. For example, if your calculator gives you 1.681792, it鈥檚 easier to work with 1.682. When rounding:
  • Focus on the digit right after your rounding point. If it's 5 or more, round up the last digit you keep.
  • In our example, since the fifth digit is 7, 1.6817 becomes 1.682.
Scientific Calculator Functions
A scientific calculator is a powerful tool that helps you carry out complex mathematics beyond simple addition or subtraction. Here are some functions that are particularly useful for calculating powers and roots.

On your calculator, look for buttons like \( y^x \) or \( \sqrt{} \) for roots and powers. These buttons allow you to input expressions like \( 8^{1/4} \) in a straightforward way.

Steps for using a scientific calculator:
  • Enter the base number, like 8, first.
  • Press the power function button, which may be labelled as \( y^x \), \( ^ \), or similar.
  • Enter the fractional exponent (0.25 for the fourth root, as \( 1/4 = 0.25 \)).
  • Hit the equals button to get your result.
Using these specific functions ensures that even complicated calculations remain manageable.

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Most popular questions from this chapter

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