Problem 8 - College Algebra

Question

Simplify each expression. $$\sqrt{289}-\sqrt{121}$$

Step-by-Step Solution

Answer

The expression simplifies to 6.

Step 1: Identify the Squares

First, we need to recognize if the numbers inside the square roots are perfect squares. The number 289 is the square of 17 because \(17 \times 17 = 289\), and the number 121 is the square of 11 because \(11 \times 11 = 121\).

Step 2: Apply the Square Root

With this information, we can now take the square root of both numbers. The square root of 289 is 17 because \( \sqrt{289} = 17\), and the square root of 121 is 11 because \( \sqrt{121} = 11\).

Step 3: Perform the Subtraction

Subtract the second square root from the first: \( 17 - 11 \). This gives the result 6.

Key Concepts

Perfect Squares

Perfect squares are numbers that are made by squaring an integer.
Essentially, if you can multiply a number by itself and get another number, that number is a perfect square. For example, the numbers like 1, 4, 9, 16, and so on, are all perfect squares because:

• 1 is 1 times 1
• 4 is 2 times 2
• 9 is 3 times 3

These numbers are important because they can be easily simplified within square roots. In our exercise, 289 is the perfect square of 17 and 121 is the perfect square of 11. Simply recognizing these numbers as perfect squares makes simplifying square root expressions much more manageable.

Square Roots

A square root is the opposite of a square. If you know a number is made by squaring another number, then you can also find what number was squared to get that. For instance, in our exercise, we found the square roots of 289 and 121, which are 17 and 11 respectively, because 17 x 17 = 289 and 11 x 11 = 121.
The square root symbol \( \sqrt{} \) is used to denote this operation. Whenever you take the square root of a perfect square, it simplifies nicely to the integer that was squared to make that perfect square. Hence, finding a square root becomes easier once you recognize whether the number inside the square root is a perfect square.

Subtraction of Square Roots

Subtracting square roots can often become simple arithmetic once the roots are simplified. Once you've simplified each square root to its simplest integer form, you can treat the problem like any regular subtraction of integers. In our problem:
Firstly, we found that:

• \( \sqrt{289} \) simplifies to 17
• \( \sqrt{121} \) simplifies to 11

Then, you simply subtract the second simplification from the first:

• 17 (from \( \sqrt{289} \)) minus 11 (from \( \sqrt{121} \)) gives us 6.

This illustrates that simplifying square roots prior to subtraction makes the process straightforward and manageable.