Problem 30 - Elementary Algebra

Question

Simplify. $$ \sqrt{b^{2}} $$

Step-by-Step Solution

Answer

\(\sqrt{b^2} = b\)

Step 1: Understand the Square Root and Squaring Relationship

Recognize that the square root function and the squaring function are inverse operations. For any non-negative number, \(a\), the square root of \(a^2\) is \(a\).

Step 2: Apply the Property to the Given Expression

Apply the property from Step 1 to the given expression, \(\sqrt{b^2}\). This simplifies to \(b\).

Key Concepts

inverse operations

In mathematics, inverse operations are pairs of operations that undo each other. This means that applying one operation and then the other returns the original value. There are many examples of inverse operations:

• Addition and subtraction
• Multiplication and division
• Squaring and square rooting

When you understand that these operations are inverses, it becomes easier to solve equations or simplify expressions. For example, if you square a number and then take the square root, you return to the original number. Recognizing and using inverse operations can simplify complex mathematical problems and make algebra easier to understand.

square root function

The square root function is a mathematical function that returns the original number when applied to a number that is squared. It's often represented by the radical symbol, \(\backslashsqrt\), followed by the number or expression. For example: \(\sqrt{16} = 4\).
The square root function has a few important properties:

• It only applies to non-negative numbers in the real number system
• The square root of a perfect square is always an integer
• It reverses the effect of squaring a number

To illustrate, if you take the square root of \(b^{2}\), the radical sign and the exponent cancel each other because they are inverse operations. So, \(\sqrt{b^{2}} = b\). Understanding this concept helps simplify mathematical expressions that involve square roots.

squaring function

The squaring function is another fundamental mathematical operation and is the inverse of the square root function. Squaring a number means multiplying it by itself. It's denoted by an exponent of 2. For instance, \(4^{2} = 16\).
Key points of the squaring function include:

• It always gives a non-negative result in the real number system
• It’s used in various mathematical contexts like geometry, algebra, and calculus
• Squaring and square rooting cancel each other out

If you square a number and then take the square root, you end up back at the original number: \(\sqrt{4^{2}} = 4\). This relationship between squaring and square rooting is crucial when simplifying or solving equations involving these operations.