Problem 83 - Fundamentals of Mathematics

Question

Use a calculator with the keys \(\sqrt{x}, y^{x}\), and \(1 / x\) to find each of the values. \(\sqrt[4]{331,776}\)

Step-by-Step Solution

Answer

The fourth root of 331,776 is 24.

Step 1: Recognize the Expression

The exercise requires us to find the value of \(\sqrt[4]{331,776}\). This means we are looking for the fourth root of 331,776.

Step 2: Use the Calculator for Fourth Root

To find the fourth root of a number, you can raise it to the power of \(\frac{1}{4}\). Use the calculator key \(y^x\) to enter this operation. Enter '331776', then press 'y^x', followed by entering '0.25', and finally press the '=' key to compute the result.

Step 3: Confirm the Calculation

When you perform the steps as described using the calculator, you should get the result 24. Therefore, \(\sqrt[4]{331,776} = 24\).

Key Concepts

Fourth Root Calculation

In mathematics, a fourth root is a number that, when raised to the power of four, gives the original number. To put it simply, if you have a number like 331,776 and want to find its fourth root, you're looking for a number that, when multiplied by itself four times, equals 331,776.Fourth roots can be calculated using the concept of powers and exponents. Specifically, if you're finding the fourth root of a number, you can express it with an exponent of \(1/4\). This works because raising a number to the power of 1/4 is the same as finding its fourth root.The concept of roots is an extension of the idea of squares and cubes. Where squaring involves multiplying a number by itself once (\(a^2\)), fourth roots involve finding a number that leads to the original number when used in power or multiplied by itself three more times beyond squaring or cubing.

Powers and Roots

Powers and roots are fundamental concepts in mathematics, used extensively for various calculations. A power refers to the number of times a number (known as the base) is multiplied by itself. For instance, if you have 3 to the power of 4, it means 3 is multiplied by itself four times: \(3^4 = 3 \times 3 \times 3 \times 3\).Roots, on the other hand, are the inverse operations of powers. They are used to find the base number when the result of its power is known. For example, if you have a number like 81, and you know it is a result of 3 raised to the power of 4 (\(3^4 = 81\)), 3 is considered the fourth root of 81.Understanding how powers and roots work is crucial for solving algebraic expressions and equations, especially those involving higher-order roots like the fourth root. By grasping these concepts, we can simplify and solve complex mathematical problems easily and accurately.

Using a Scientific Calculator

Computing higher order roots like the fourth root can be easily done using a scientific calculator. These calculators come equipped with various functions to simplify complex calculations by just pressing a few keys.To compute a fourth root using a scientific calculator, you need to use the exponent key, often labeled as \('y^x‘\) or a similar symbol. Here's how you can do it:

• Enter the number for which you want to find the fourth root; for example, 331,776.
• Press the \('y^x‘\) key to indicate you want to raise the number to a power.
• Enter \(.25\) since raising a number to the power of 0.25 is equivalent to taking the fourth root.
• Finally, press the '=' key to get the result.

You should see 24 as the result on your calculator. It's always helpful to note that the \('y^x‘\) operation allows finding any root beyond just square or cube roots, extending the calculator's utility for more complex math problems.