Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{289}-\sqrt{121}\)

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Simplify the First Square Root

The first step is to simplify the square root of 289. Recall that finding the square root means identifying a number that, when multiplied by itself, equals 289. In this case, the number is 17 since \(17 \times 17 = 289\). Therefore, we have \(\sqrt{289} = 17\).

Step 2: Simplify the Second Square Root

Next, simplify the square root of 121. Similarly, find a number that, when multiplied by itself, equals 121. This number is 11 because \(11 \times 11 = 121\). Thus, \(\sqrt{121} = 11\).

Step 3: Calculate the Difference

Now, subtract the simplified square root of 121 from the simplified square root of 289. The operation is as follows: \(17 - 11 = 6\).

Key Concepts

Square Roots

The concept of square roots is fundamental in mathematics. A square root of a number is a value that yields the original number when multiplied by itself. In simple terms, if you have a number, say 9, and you ask "what number multiplied by itself makes 9?", the answer is 3. So, the square root of 9 is 3, because \(3 \times 3 = 9\).

This operation is commonly represented by the radical symbol \(\sqrt{}\). For example, \(\sqrt{16}\) equals 4, since \(4 \times 4 = 16\). Square roots can only be accurately simplified when dealing with perfect squares - numbers that are the squares of integers. If they're not perfect squares, we often leave them in their root form or use approximations. Understanding square roots help us simplify expressions and solve equations efficiently.

Arithmetic Operations

Arithmetic operations in mathematics include basic processes like addition, subtraction, multiplication, and division. When dealing with square roots, arithmetic operations are applied in the same way as with regular numbers. However, specific care must be taken during calculations.

When you perform arithmetic operations with square roots:

• Ensure that each root is first simplified, if possible.
• Square roots of whole numbers can become integers if the number is a perfect square.
• Subtraction and addition involve simplified values just as ordinary arithmetic.

In our example exercise, we have \(\sqrt{289} - \sqrt{121}\). After simplifying each square root, we perform the arithmetic operation of subtraction: \(17 - 11\), which results in 6.

Applying arithmetic operations correctly ensures that you achieve the exact and simplified outcome.

Simplification Process

The simplification process involves reducing expressions to their simplest form, which can often help in providing clarity and ease of computation. This involves breaking down complex expressions into more manageable ones or finding the exact equivalent value.

When simplifying square roots:

• Identify if the number under the root is a perfect square. If so, substitute the square root with the integer that squares to form the original number.
• Use multiplication rules to manage roots that can be combined or factored.
• Subtract or add resolved integers rather than dealing directly with roots when possible.

For instance, in our exercise with \(\sqrt{289} - \sqrt{121}\), each square root was simplified into an integer, \(17\) and \(11\), respectively, before subtracting them to get a straightforward integer result of 6. Simplification not only makes computations easier but also helps in solving mathematical problems more effectively.