Problem 15 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \frac{\sqrt{a^{2}+6 a+9}}{\sqrt{a+3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{\sqrt{a^2+6a+9}}{\sqrt{a+3}}\), and state its domain. Answer: The simplified expression is \(\frac{a+3}{\sqrt{a+3}}\) for \(a \ge -3\).

Step 1: Identify the given expression

We are given the expression: $$ \frac{\sqrt{a^2+6a+9}}{\sqrt{a+3}} $$

Step 2: Factor the numerator

We know that \((a+b)^2 = a^2 + 2ab + b^2\). In our case, we can factor \(a^2+6a+9\) as \((a+3)^2\). So the expression becomes: $$ \frac{\sqrt{(a+3)^2}}{\sqrt{a+3}} $$

Step 3: Take the square root of numerator

Using the property \(\sqrt{x^2}=|x|\), we have: $$ \frac{|a+3|}{\sqrt{a+3}} $$

Step 4: Consider the domain of the expression

Before we simplify further, we need to consider the domain of the expression to avoid division by zero or square root of a negative number. We need \(a+3\ge 0\). Thus, we require \(a\ge-3\).

Step 5: Simplify the expression

Since we know \(a+3 \ge 0\), we can eliminate the absolute value sign, and then simplify the expression as follows: $$ \frac{a+3}{\sqrt{a+3}} $$ The expression is now simplified, and the final result is: $$ \frac{a+3}{\sqrt{a+3}} \quad \text{for} \quad a \ge -3 $$

Key Concepts

Factoring Polynomials

Polynomials are expressions that consist of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents. Factoring, a key skill in algebra, is the process of breaking down a polynomial into simpler components - known as factors - that, when multiplied together, give back the original polynomial. It's very similar to factoring a number into its prime components; just as 15 can be factored into 3 x 5, a polynomial such as \(x^2+5x+6\) can be broken down into \((x+2)(x+3)\).

When simplifying algebraic expressions, particularly before performing operations like division, factoring polynomials can be incredibly useful. For instance, the square of a binomial, like \((a+3)^2\), is a polynomial that can be factored into the product of identical binomials. Recognizing common factoring patterns, such as the difference of squares or perfect square trinomials, can simplify many algebraic expressions. In our textbook problem, factoring transformed \(a^2+6a+9\) into \((a+3)^2\), streamlining the rest of the simplification.

Square Root Properties

The square root of a number represents a value, which, when multiplied by itself, yields the original number. The properties of square roots are applied in simplifying expressions like the one in our exercise. An important property is \(\sqrt{x^2}=|x|\), where the absolute value ensures the result is non-negative because a square root by definition cannot be negative.

Another key consideration is that the square root is a radical operation, and laws of exponents play a significant role. For example, \(\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}\) and \(\left(\sqrt{x}\right)^2=x\). These properties allow us to manipulate and simplify expressions under the radical sign. It's noteworthy that the square root of a squared expression simplifies to the absolute value of the expression within the square, which was applied when simplifying \(\sqrt{(a+3)^2}\) to \(|a+3|\) in our problem.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For example, \(|-4|=4\) and \(|4|=4\). In an algebraic context, the absolute value function turns any negative input into its positive counterpart, and it leaves a positive input unchanged.

Understanding the concept of absolute value is critical when dealing with square roots because, as mentioned, the square root of a square is the absolute value of the base. This is to ensure that the square root of a square yields a non-negative result. In our example, after factoring the polynomial under the radical sign, we get \(\sqrt{(a+3)^2}\) which simplifies to \(|a+3|\), demonstrating the use of absolute value in dealing with square roots.

Domain of an Expression

In mathematics, the domain of an expression is the set of all values that can be used as inputs without causing errors in calculation, such as division by zero or taking the square root of a negative number in the real number system. Determining the domain is a critical step when simplifying expressions, especially those involving radicals or fractions.

In our problem, we are working with the expression \(\sqrt{a+3}\), which requires us to restrict the domain to ensure the value under the square root is non-negative, since the square root of a negative number is not real. We determine that \(a+3\) must be greater than or equal to zero, leading to the restriction on the variable: \(a \geq -3\). With this understanding, we can confidently proceed with simplifications and know that the final expression will be valid for all permissible values of the variable within the defined domain.