Problem 292 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \sqrt{169} $$

Step-by-Step Solution

Answer

13

Step 1: Identify the Expression

The expression to simplify is \( \sqrt{169} \). This is a square root operation, which asks for a number that, when multiplied by itself, gives 169.

Step 2: Recognize Perfect Squares

Recall that 169 is a perfect square because there exists an integer which, when squared, results in 169. This integer is 13.

Step 3: Simplify the Square Root

The square root of a perfect square is the integer that was squared to produce it. Therefore, \( \sqrt{169} = 13 \).

Key Concepts

Square Roots

Let's start with what a square root is. A square root asks for a number that, when multiplied by itself, equals the given number. For example, in the problem, you are given the square root of 169: \[ \sqrt{169} \].
Here, you need to find the number that, when squared (multiplied by itself), equals 169.
Square roots are fundamental in math because they help in various operations including statistics, geometry, and algebra.
To simplify a square root, it helps to know about perfect squares, which we will discuss next.

Perfect Squares

Understanding perfect squares is crucial to simplifying square roots. A perfect square is a number that is the product of an integer multiplied by itself.
Some common perfect squares include:

• \[ 1^2 = 1 \]
• \[ 2^2 = 4 \]
• \[ 3^2 = 9 \]
• \[ 4^2 = 16 \]
• \[ 5^2 = 25 \]
• And so on...

From this, we can see that \[ 13^2 = 169 \]. This signifies that 169 is a perfect square. Knowing this, we can simplify the square root of 169 to 13.

Integer Multiplication

Finally, let's talk about integer multiplication, since it plays a big role in understanding square roots. Integer multiplication is simply multiplying whole numbers.
For instance, when you multiply 13 by itself (\[ 13 \times 13 \]), you get 169.
So, when you're simplifying \[ \sqrt{169} \], you're essentially asking: what number times itself gives 169? The answer is 13. Always remember:

• Multiplying an integer by itself results in a perfect square.
• When simplifying a square root of a perfect square, you're looking for the number that was squared.