Problem 84 - Fundamentals of Mathematics

Question

Use a calculator with the keys \(\sqrt{x}, y^{x}\), and \(1 / x\) to find each of the values. \(\sqrt[8]{5,764,801}\)

Step-by-Step Solution

Answer

The eighth root of 5,764,801 is 19.

Step 1: Understand the Expression

The expression \(\sqrt[8]{5,764,801}\) represents the eighth root of 5,764,801. This means that you need to find a number which, when raised to the power of 8, equals 5,764,801.

Step 2: Use the 1/x key

To find the eighth root using a calculator, you can first press the 1/x key to transform \(8\) into its reciprocal since the eighth root of a number is the same as raising that number to the power of \(\frac{1}{8}\).

Step 3: Use the y^x key

Next, input 5,764,801 and use the \(y^x\) key on your calculator. After this, press the result of Step 2 to compute \(5,764,801^{\frac{1}{8}}\).

Step 4: Read the Result

Evaluate the final result from your calculator. It will give you the value of \(5,764,801^{\frac{1}{8}}\).

Key Concepts

Eighth Root Calculation

The eighth root of a number is a special type of root. It's the value that, when we multiply it by itself eight times, gives us the original number. For instance, if we want to find the eighth root of 5,764,801, we're searching for a number that, raised to the power of 8, equals 5,764,801.

In calculations, the eighth root of a number can be represented as a fraction exponent:

• Mathematically, this is equivalent to raising the number to the power of \( \frac{1}{8} \).
• For example, the expression \( \sqrt[8]{5,764,801} \) is the same as \( 5,764,801^{\frac{1}{8}} \).

Calculators often don't have specific root keys for higher roots like the eighth. This transformation into a fractional exponent enables us to use a scientific calculator with the exponentiation capabilities to get our answer.

Reciprocal Function

The reciprocal function is essential in transforming root calculations into power calculations, particularly when using a calculator. The reciprocal of a number is simply one divided by that number.

For example:

• The reciprocal of 8 is \( \frac{1}{8} \).
• If you have a calculator, you often use the \(1/x\) key to find this reciprocal.

Using the reciprocal helps in situations like our eighth root problem, where a direct root operation might not be feasible.

• By using the \(1/8\) exponent, we convert the eighth root into a format that a calculator can handle using its power function.

This transformation allows us to efficiently compute roots beyond simple square or cube roots, broadening the functionality of our calculators.

Exponentiation Process

Once we have transformed the root into a fractional exponent using the reciprocal function, the next step is exponentiation. Exponentiation involves raising a number to the power of another, using the calculator's \(y^x\) or similar function.

Specifically, to find the eighth root of 5,764,801:

• We first determined \(\frac{1}{8}\) as the fraction exponent.
• Next, on a calculator, input 5,764,801.
• Use the function, often labeled \(y^x\), to raise this number to the calculated power of \(\frac{1}{8}\).

This operation allows the calculator to determine the eighth root efficiently by computing the number multiplied by itself a fractional number of times, resulting in the original number or its root. Understanding this connection between roots and exponents is key for solving many mathematical problems with calculators.