Problem 106 - Algebra A Combined Function

Question

Use a calculator to write a four-decimal-place approximation of each number. $$ 18^{3 / 5} $$

Step-by-Step Solution

Answer

The four-decimal-place approximation of \(18^{3/5}\) is 6.2232.

Step 1: Understand the Expression

The expression given is \(18^{3/5}\). This means 18 raised to the power of \(\frac{3}{5}\). The objective is to find its approximate value using a calculator.

Step 2: Use the Calculator

Input the expression into a calculator: Use the exponentiation function, typically marked as \(^\wedge\) or \(x^y\) depending on your calculator model. Enter 18 as the base, followed by the power \(\frac{3}{5}\) or 0.6.

Step 3: Get the Result

Upon entering the expression correctly, the calculator will provide an approximation of the result.

Step 4: Round to Four Decimal Places

The calculator gives a result. Round this result to four decimal places for accuracy.

Key Concepts

Rational Exponents

Rational exponents may seem tricky initially, but they're just a different way to write roots and powers. When you see an exponent like \(\frac{3}{5}\), it implies two operations are happening. The fraction is split into a numerator and a denominator. Here’s how to interpret it:

• Numerator (3): This is the power you raise the number to.
• Denominator (5): This represents the root you take afterwards.

To solve \(18^{3/5}\), it can be understood as taking the fifth root of 18 and then cubing the result, or vice versa. Both methods will yield the same answer, thanks to the properties of exponents that let us multiply powers one step at a time. This is a powerful concept that makes complex mathematical calculations more manageable!

Calculator Usage

Using a calculator can be a lifesaver in getting decimal approximations for expressions like \(18^{3/5}\). Modern calculators have specific functions that allow you to input rational exponents directly. Here’s a simple guide to ensure accuracy:

• Locate the exponentiation button: This is usually marked as \(^\wedge\) or \(x^y\).
• Input the base number: Enter 18 into your calculator.
• Enter the exponent: Type in the exponent, which is \(\frac{3}{5}\). Just input 0.6 if your calculator doesn't support fractional exponents.
• Press the enter or equals button to execute the calculation.

Not only does this provide an accurate result, but it also helps students focus on understanding the process rather than manual calculations. Plus, modern calculators often have features to recall past calculations, handy for verifying your steps if the answer seems off.

Decimal Approximation

Decimal approximation is the process of rounding a value to a specific number of decimal places. It’s essential for practical purposes, particularly when dealing with irrational numbers. Suppose you've calculated the expression \(18^{3/5}\) and received a lengthy decimal output.
Here's how to round correctly to four decimal places:

• Look at the fifth decimal place.
• If it's 5 or more, round the fourth decimal place up.
• If it's less than 5, keep the fourth decimal place as is.

For example, if your calculator shows 7.8786431, you round it to 7.8786 based on these rules.
This practice ensures the answers are tidy and fit the precision required for your task at hand. Such accuracy is often essential in fields like engineering, science, and finance, where even small discrepancies can have large impacts.