Problem 37 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{49 p} $$

Step-by-Step Solution

Answer

The simplified form is \(7\sqrt{p}\).

Step 1: Simplify the square root of a perfect square

The first step is to recognize any perfect squares under the square root. In the expression \(\sqrt{49p}\), we see that 49 is a perfect square because \(49 = 7^2\). Therefore, \(\sqrt{49} = 7\).

Step 2: Recombine terms

After simplifying the perfect square, combine the terms. Therefore, \(\sqrt{49p} = 7\sqrt{p}\). Since \(p\) is not a perfect square, the expression cannot be simplified further with respect to \(p\).

Key Concepts

Perfect Squares

A perfect square is a number that is the result of squaring an integer. For example, when you multiply an integer by itself, the result is called a perfect square. This is an important concept when dealing with square roots because recognizing perfect squares can significantly simplify your calculations.

• For instance, 49 is a perfect square because it is the result of 7 times 7.
• Similarly, numbers like 4, 9, 16, and 25 are all perfect squares.

Knowing about perfect squares is useful in algebra because it allows us to simplify expressions that involve square roots. It also helps us in factoring numbers, which is often needed to solve algebraic equations.

Square Roots

The square root is essentially the opposite of squaring a number. When we take the square root of a number, we are asking what number squared gives us the original number. This concept is most frequently used in simplifying algebraic expressions, especially when dealing with perfect squares.

• The square root of 49, a perfect square, is 7 because 7 times itself equals 49.
• If the number or expression inside the square root isn't a perfect square, such as a variable like \(p\), the root cannot be simplified further.

Square roots are visually represented by the radical symbol \(\sqrt{}\). When simplifying expressions, identify and extract any perfect squares under the radical to simplify calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. Simplifying these expressions often involves reducing the expression to a more manageable form.

• They can become complex when they include square roots, such as in the expression \(\sqrt{49p}\).
• The goal is to simplify them by breaking down numbers into factors, recognizing perfect squares, or performing operations like addition and subtraction.

In our example \(\sqrt{49p}\), this means recognizing that 49 is a perfect square and simplifying it to 7, leaving us with \(7\sqrt{p}\). This is a crucial step in simplifying algebraic expressions as it often makes further calculations much easier to handle.