Problem 37 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Perfect Squares

Examine the expression under the square root: \( \sqrt{49p} \). Note that 49 is a perfect square because it equals \(7^2\).

Step 2: Apply the Square Root Property

Using the property \( \sqrt{a^2} = a \), simplify the square root of a perfect square. Thus, \( \sqrt{49} = 7 \).

Step 3: Simplify the Expression

Substitute the simplified square root back into the expression: \( \sqrt{49p} = 7 \sqrt{p} \). This is the simplified form of the original expression.

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer by itself. For instance, the number 49 is a perfect square because it is equal to 7 times 7, or 7 squared (\(7^2\)). The concept of perfect squares is very useful when simplifying square root expressions.

Recognizing perfect squares can make your calculations much easier. Whenever you see a number under a square root, check if it is a perfect square:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all examples of perfect squares.
• This means that the square roots of these numbers are integers (1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 respectively).

Identifying and working with perfect squares helps streamline the process of simplifying expressions, making it less complicated and time-consuming.

Square Root Property

The square root property is a handy mathematical tool used in simplifying expressions that involve square roots. It is based on the principle that the square root of any squared number returns the base number.

In mathematical terms, this property is expressed as \(\sqrt{a^2} = a\). Essentially, if you take the square root of a perfect square, you get back to the original whole number, which is very helpful in simplification.

• For example, when simplifying \(\sqrt{49}\), since 49 is \(7^2\), the square root is 7.
• If an expression includes variables, such as \(\sqrt{x^2}\), you can also use this property to simplify it to \(x\).

Using the square root property enables us to simplify expressions more efficiently, especially when dealing with perfect squares. This simplification is a key step in ensuring that algebraic expressions are easier to manage and understand.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations (such as addition, subtraction, and multiplication). They can become quite complex, but simplifying them can make them easier to deal with.

Simplifying algebraic expressions often involves removing complexity from the expression, such as reducing fractions, combining like terms, and applying mathematical properties—like factoring perfect squares and using the square root property.

• Consider an expression like \(\sqrt{49p}\). With the perfect square 49, the simplification begins by recognizing this and using the square root property.
• Then, you tackle the expression by reducing it to its simplest form, such as \(7\sqrt{p}\).

Mastering these simplification techniques allows students to solve more complicated problems with less frustration, providing a practical skill for various mathematical applications.