Problem 15 - College Algebra

Question

Simplify each expression. $$\sqrt{169}+\sqrt{144}$$

Step-by-Step Solution

Answer

The simplified expression is 25.

Step 1: Identify the Square Roots

The expression we need to simplify is \( \sqrt{169} + \sqrt{144} \). Start by determining the square roots of both numbers. Recognize that 169 and 144 are perfect squares.

Step 2: Calculate Each Square Root

Calculate \( \sqrt{169} \). Since 13 squared (\( 13^2 \)) equals 169, \( \sqrt{169} = 13 \). Similarly, calculate \( \sqrt{144} \). Since 12 squared (\( 12^2 \)) equals 144, \( \sqrt{144} = 12 \).

Step 3: Add the Results

Now that we know \( \sqrt{169} = 13 \) and \( \sqrt{144} = 12 \), add these values together: \( 13 + 12 = 25 \).

Key Concepts

Square Roots

Square roots are an essential concept in mathematics. They represent a value that, when multiplied by itself, gives the original number. For example, the square root of 169 is 13 because multiplying 13 by itself gives you 169.

• A square root is represented by the symbol \( \sqrt{} \).
• Finding the square root of a number is essentially the opposite of squaring it.

When dealing with square roots, it's crucial to determine whether the number under the square root symbol is a perfect square. Perfect squares are numbers like 4, 9, 16, and 25, which can be expressed as the square of a whole number.

Perfect Squares

Perfect squares are numbers that result from squaring integers. Observing numbers like 169 and 144, which appear in our original expression \( \sqrt{169} + \sqrt{144} \), you notice they both are perfect squares. Here's why:

• 169 is a perfect square because it is \( 13^2 \), where 13 is an integer.
• 144 is a perfect square because it is \( 12^2 \), where 12 is an integer.

Recognizing perfect squares is beneficial because it allows you to easily calculate the square roots of these numbers, simplifying the process of solving expressions.

Addition of Radicals

The addition of radicals incorporates the basic principle of combining like terms. However, with square roots (or radicals), you first need to simplify each radical if possible by determining their square roots, especially when dealing with perfect squares.For example, to solve the expression \( \sqrt{169} + \sqrt{144} \), firstly calculate the square roots of each number:

• \( \sqrt{169} = 13 \)
• \( \sqrt{144} = 12 \)

Once simplified, the expression becomes a simple addition problem: \( 13 + 12 = 25 \). This stepwise simplification shows how radicals can become a straightforward addition when each term has been simplified first.