Problem 33 - Algebra 1

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$\sqrt{13}$$

Step-by-Step Solution

Answer

The value of \( \sqrt{13} \), rounded to the nearest hundredth, is approximately 3.61.

Step 1: Understanding the Square Root Function

The square root function, denoted as \( \sqrt{x} \), provides the value which when squared equals the input \(x\). Here the input is 13.

Step 2: Compute the Square Root

Use the square root function on a calculator to compute \( \sqrt{13} \). The result is a non-terminating decimal.

Step 3: Rounding to the Nearest Hundredth

Finally, round the computed value to the nearest hundredth. This is the standard convention when an exact value isn't possible.

Key Concepts

Approximating Square Roots

When you encounter a square root like \( \sqrt{13} \), the goal is often to approximate its value when an exact one isn't readily available. This means finding a value that is close enough to the actual square root. Approximations are important in mathematics when dealing with irrational numbers—those that cannot be precisely expressed by a finite sequence of digits.

Knowing how to approximate helps us make useful and practical calculations even with complex numbers. For instance, while the exact square root of 13 is unavailable, approximate values, such as around 3.6055512, can serve well in everyday scenarios where high precision isn't critical.

Here's a quick way to remember steps for approximation:

• Find the closest perfect squares. In this case, \(3^2 = 9\) and \(4^2 = 16\), so \(\sqrt{13}\) lies between 3 and 4.
• Use a calculator or estimation to hone in on the approximate decimal value.

Using these strategies ensures you have a helpful approximation for practical applications.

Rounding Non-Terminating Decimals

Non-terminating decimals are decimals that go on forever without repeating. When computing \( \sqrt{13} \), the resulting decimal is non-terminating and notably approximately 3.605551275. Since we cannot write all the digits, we often round to a manageable number of decimal places.

Rounding helps simplify numbers for easier use. It's like cutting off the decimal tail of a number to fit our needs better. In the context of \( \sqrt{13} \), we might round it to the nearest hundredth (two decimal places) for simplicity. Here’s the rounded form: 3.61.

Rounding follows a few simple rules:

• Look at the digit immediately after your desired decimal place. If it’s 5 or more, round up.
• If it’s less than 5, keep your rounding point unchanged.

Utilizing rounding allows you to handle non-terminating decimals effectively, making them practical for everyday calculation tasks.

Understanding Non-Terminating Decimals

Dealing with non-terminating decimals can sometimes be overwhelming. A non-terminating decimal continues indefinitely without settling into a repeating pattern. For example, when we calculate \( \sqrt{13} \), we get approximately 3.605551275, a decimal that stretches on and does not terminate.

In math, these decimals are often the result of an irrational number. Most square roots of numbers that are not perfect squares end up being irrational, leading to non-terminating decimals. This can make exact representation in calculations challenging.

Here's how to approach non-terminating decimals:

• Recognize that not all numbers can be cleanly framed as fractions or simple decimals.
• Use rounding or truncation to give a simpler, usable form, especially in practical applications where precision is less crucial.

Understanding and manipulating non-terminating decimals is crucial when working with square roots and other calculations involving irrational numbers.