Question

Use a calculator to approximate the number. (Round to three decimal places.)\(3.75^{-1 / 2}\)

Short answer

The approximation of \(3.75^{-1 / 2}\) to three decimal places is 0.577

Step-by-step solution

Calculate the Reciprocal

Calculate the reciprocal of 3.75, which is given by \(1/3.75\). Use a calculator to divide 1 by 3.75 to determine the reciprocal to three decimal places.

Take the Square Root

Now compute the square root of the reciprocal found in step 1. In other words, calculate \(\sqrt{1/3.75}\) using a calculator.

Round the Result

Round the resulting value from Step 2 to three decimal places using standard rules of rounding.

Key concepts

Reciprocal

In mathematics, the reciprocal of a number is simply one divided by that number. It's like flipping the fraction. For example, the reciprocal of 5 is \( \frac{1}{5} \). A reciprocal can help us in various calculations, including exponentiation. If you have a number like 3.75, to find its reciprocal, you do the following:

• Take the number: 3.75.
• Divide 1 by this number: \( \frac{1}{3.75} \).
• Use a calculator to find an approximate value if needed.

In our case, we calculate this using a calculator for precision, which is crucial for accurate results in subsequent steps.

Square Root

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). The symbol used for square root is \( \sqrt{} \). When dealing with reciprocal and square root together, like in the problem \( \sqrt{\frac{1}{3.75}} \), follow these steps:

• Start with the reciprocal you've calculated.
• Use a calculator to accurately find the square root.
• Make sure the square root operation is clear in the calculation.

Finding the square root of a reciprocal might seem complex, but it's straightforward when approached step-by-step with the help of a calculator.

Rounding

Rounding is the process of simplifying a number while maintaining its value close to what it initially was. When asked to round to three decimal places, like in our exercise, focus on the first four digits of the result:

• Look at the fourth decimal place.
• If it's 5 or more, increase the third decimal by 1.
• If it's less than 5, keep the third decimal as it is.

For example, if your calculator displays 0.514288, rounding to three decimal places would give you 0.514. Mastering rounding is essential for precision in mathematical work.

Calculator Use

Calculators are helpful tools for performing accurate and quick calculations, especially when dealing with exponentials, reciprocals, and square roots. When using a calculator:

• Ensure the correct mode (degree/radian or decimal/fraction) is set for your calculation.
• Double-check if you've entered the numbers correctly to avoid errors.
• For complex expressions, follow the order of operations: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS).

For exercises that demand precision, such as finding square roots and reciprocals, the calculator helps to avoid manual errors and ensures the result is rounded appropriately to the required decimal places.