Problem 98 - Algebra and Trigonometry

Question

\(\sqrt{89}\)

Step-by-Step Solution

Answer

√89 ≈ 9.434

Step 1 - Understand the Problem

The goal is to simplify \(\text{√89}\). Since 89 is not a perfect square, the number is irrational and cannot be simplified further.

Step 2 - Use a Calculator

To get an approximate value for \(\text{√89}\), use a calculator.

Step 3 - Evaluate the Expression

Enter \(\text{√89}\) into the calculator, which gives approximately 9.434.

Key Concepts

irrational numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as the ratio of two integers. They have non-repeating, non-terminating decimal expansions.
For example, \(\text{√2}\), \(\text{π}\), and the number in our exercise \(\text{√89}\) are all irrational numbers. This is because there are no two integers you can divide to get these values exactly.
When simplifying square roots, if the number under the square root (called the radicand) is not a perfect square, the result is irrational.
For instance:
\[\text{√2}=1.4142135…\]
Similarly:
\[\text{√89}=9.434…\] There is no exact fraction to represent these values.
Remembering the characteristics of irrational numbers can help you solve and understand problems like our exercise today.

perfect squares

A perfect square is a number that is the square of an integer. Examples include 1, 4, 9, 16, 25, 36, and so on.
Perfect squares are important because they make simplifying square roots straightforward. For example, \(\text{√16}=4\) because 4 multiplied by itself equals 16.
In our exercise, we are dealing with \(\text{√89}\). Here, 89 is not a perfect square because there is no whole number that, when multiplied by itself, equals 89.
Identifying if a number is a perfect square helps determine whether you can simplify the square root to an integer or if it remains an irrational number.
When working with square roots, always check if the radicand is a perfect square first. If it is not, you will likely end up with an irrational number like \(\text{√89}\).

using a calculator

Calculators are an essential tool for finding approximate values of square roots, especially when dealing with irrational numbers.
To find the square root of a number using a calculator, follow these simple steps: 1. Turn on the calculator. 2. Press the square root (√) button. 3. Enter the number you want the square root of—in this case, 89. 4. Press = or Ansr (answer).
The display will show an approximate value. For \(\text{√89}\), the result will be roughly 9.434.
This approximation is useful when you need a numerical value rather than just the symbolic form. While the original number remains irrational, having a decimal approximation can be beneficial for practical applications and further calculations.