Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$\pm \sqrt{0.75}$$

Short answer

The solutions are approximately \(\pm 0.87\).

Step-by-step solution

Calculate the square root of 0.75

Firstly, compute the square root of 0.75. The square root function in a calculator will come in handy for this calculation.

Give positive and negative solutions

Because of the \(\pm\) symbol, there will be two answers for this expression: one for \(\sqrt{0.75}\) (the positive square root) and -\(\sqrt{0.75}\) (the negative square root). So after finding the square root, simply give the positive and negative of the result. Remember to round your solutions to the nearest hundredth as stated in the problem if not an exact value.

Key concepts

Approximating Decimals

When dealing with square roots of decimals such as \(\pm \sqrt{0.75}\), it's not always possible to find an exact value. Therefore, we approximate the decimal value to make it understandable and usable in calculations. Especially in applications, approximations allow us to work with a practical number rather than an infinite decimal.

To approximate the decimal value of a square root like \(\sqrt{0.75}\), use a calculator to find the square root and then round the result. While the calculator gives a close approximation, the process of rounding brings the value to a fixed number of decimal places. For example, when rounding to the nearest hundredth, you'd want the result to have two digits after the decimal point.

Why is approximating important? In real-world situations, working with infinite decimal expansions is impractical, so approximations help in making sensible predictions and measurements.

Positive and Negative Square Roots

When you see an expression like \(\pm \sqrt{0.75}\), it's important to recognize that square roots inherently have two solutions: one positive and one negative. This is because squaring either a positive or negative number results in a positive value. For instance, both \( 3^2 \) and \( (-3)^2 \) equal 9.

To find the positive square root of a number, use the square root function on your calculator. The positive square root is the principal root and is what you initially get from most calculators. To find the negative square root, simply take the opposite of the positive square root number. It’s crucial to include both answers to fully solve the problem, as the presence of \(\pm\) denotes that both roots are sought.

Remembering the Principle of Square Roots

Always start with the principal (positive) square root when solving equations, but keep in mind a negative root can also exist, and its inclusion is necessary when an equation specifies the \(\pm\) symbol.

Rounding Decimals

Rounding decimals is a fundamental skill that helps us simplify numbers to make them easier to work with or understand. When a problem instructs you to round to the nearest hundredth, this means that you need to look at the third decimal place of the number.

If the third decimal place is 5 or greater, the second digit is increased by one. If it’s less than 5, the second digit stays the same, and all following digits are dropped. For instance, \(\sqrt{0.75}\) might be approximately 0.86603 on your calculator, rounding to the nearest hundredth would give you 0.87.

Why Rounding is Used

Rounding makes complex decimals manageable and standardizes numbers across various fields like finance and engineering. It's a way of tidying up figures or estimates to convey information efficiently without the need for excessive and often non-essential precision.