Lesson 3: Simplifying Algebraic Expressions II (Part 2)

MA158 — Spring 2026

Radical Expressions

In this section, we explore properties and techniques for simplifying expressions involving square roots and other radicals. Understanding these rules allows us to manipulate radical expressions algebraically and simplify complex expressions.

Radical Rules

For nonnegative values of \(a\) and \(b\) (where \(b \neq 0\)):

\[\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\] \[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]

Warning: Square Root Does Not Distribute Over Addition

\(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\)

That is, the square root does not distribute over addition or subtraction. For example, the expression \(\sqrt{x^2+4}\) cannot be further simplified by breaking it apart.

Simplified Form for Radical Expressions

The most simplified version of an expression with square roots should not contain:

(1) perfect squares under the radical

(2) radicals in the denominators

When radicals appear in the denominator, we eliminate them by multiplying by the conjugate. The conjugate of \(a + \sqrt{b}\) is \(a - \sqrt{b}\), and vice versa.

Note on Radicals in Denominators

For the purpose of this class, having a radical in the denominator may be okay in some contexts. However, it is important to know how to remove radicals from denominators when needed, as this is a standard algebraic technique.

Simplifying Radical Expressions

Example 1: Simplifying Radical Expressions

(a) Simplify \(\sqrt{1200}\)

We begin by factoring 1200 to identify perfect squares:

\[\sqrt{1200} = \sqrt{12 \cdot 100} = \sqrt{12} \cdot \sqrt{100} = \sqrt{12} \cdot 10\]

We continue simplifying \(\sqrt{12}\):

\[\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\]

Therefore:

\[\sqrt{1200} = 10 \cdot 2\sqrt{3} = 20\sqrt{3}\]

(b) Simplify \(\sqrt{\frac{64}{3}}\)

Using the quotient rule for radicals:

\[\sqrt{\frac{64}{3}} = \frac{\sqrt{64}}{\sqrt{3}} = \frac{8}{\sqrt{3}}\]

To rationalize the denominator, we multiply both numerator and denominator by \(\sqrt{3}\):

\[= \frac{8}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{3}\]

Rationalizing Denominators with Conjugates

When a denominator contains a sum or difference involving a radical, we use the conjugate to rationalize it. This technique relies on the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\).

Side Note: Difference of Squares with Radicals

When we multiply conjugates involving radicals, we use the pattern:

\[(a - b)(a + b) = a^2 - b^2\]

For example:

\[(x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3\]

Example 1 (continued): Rationalizing with Conjugates

(c) Simplify \(\displaystyle\frac{4}{x + \sqrt{3}} \cdot \frac{x - \sqrt{3}}{x - \sqrt{3}}\)

We multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{4}{x + \sqrt{3}} \cdot \frac{x - \sqrt{3}}{x - \sqrt{3}} = \frac{4(x - \sqrt{3})}{(x + \sqrt{3})(x - \sqrt{3})} \]

Using the difference of squares formula in the denominator:

\[= \frac{4(x - \sqrt{3})}{x^2 - 3}\]

(d) Simplify \(\displaystyle\frac{5 - x}{\sqrt{x} + \sqrt{5}} \cdot \frac{\sqrt{x} - \sqrt{5}}{\sqrt{x} - \sqrt{5}}\)

We multiply by the conjugate of the denominator:

\[ \frac{5 - x}{\sqrt{x} + \sqrt{5}} \cdot \frac{\sqrt{x} - \sqrt{5}}{\sqrt{x} - \sqrt{5}} = \frac{(5-x)(\sqrt{x} - \sqrt{5})}{(\sqrt{x} + \sqrt{5})(\sqrt{x} - \sqrt{5})} \]

The denominator simplifies using the difference of squares:

\[(\sqrt{x})^2 - (\sqrt{5})^2 = x - 5\]

So we have:

\[= \frac{(5-x)(\sqrt{x} - \sqrt{5})}{x - 5}\]

Notice that \(5 - x = -(x - 5)\), so:

\[ = \frac{-(x-5)(\sqrt{x} - \sqrt{5})}{x - 5} = -(\sqrt{x} - \sqrt{5}) = -\sqrt{x} + \sqrt{5} = \sqrt{5} - \sqrt{x} \]

Understanding Square Roots of Squared Expressions

Square Root of a Squared Variable

When \(a \geq 0\), we have \(\sqrt{a^2} = a\).

But what happens if \(a < 0\)? For example, if \(a = -4\), then:

\[(\sqrt{a})^2 = \sqrt{(-4)^2} = \sqrt{16} = 4\]

In general, for any \(x\) (positive or negative):

\[\sqrt{x^2} = |x|\]

where \(|x|\) denotes the absolute value of \(x\).

Example 2: Simplifying Expressions (Assume \(x\) is Positive)

(a) Simplify \(\sqrt{\frac{24x^3}{6x}}\)

First, simplify the fraction under the radical:

\[ \sqrt{\frac{24x^3}{6x}} = \sqrt{\frac{24}{6} \cdot \frac{x^3}{x}} = \sqrt{4 \cdot x^2} = \sqrt{4} \cdot \sqrt{x^2} = 2x \]

(b) Simplify \(\sqrt{\frac{11x}{4x^3}}\)

Simplify the fraction under the radical:

\[ \sqrt{\frac{11x}{4x^3}} = \sqrt{\frac{11}{4x^2}} = \frac{\sqrt{11}}{\sqrt{4x^2}} = \frac{\sqrt{11}}{2x} \]

Example 3: More Complex Simplifications

(a) Simplify \(\displaystyle\frac{3}{\sqrt{x+1} - 2}\)

Multiply the numerator and denominator by the conjugate \(\sqrt{x+1} + 2\):

\[ \frac{3}{\sqrt{x+1} - 2} \cdot \frac{\sqrt{x+1} + 2}{\sqrt{x+1} + 2} = \frac{3(\sqrt{x+1} + 2)}{(\sqrt{x+1})^2 - 4} \] \[= \frac{3(\sqrt{x+1} + 2)}{x + 1 - 4} = \frac{3(\sqrt{x+1} + 2)}{x - 3}\]

(b) Simplify \(\displaystyle\frac{4}{\sqrt{x-1} + \sqrt{x+2}}\)

Multiply by the conjugate \(\sqrt{x-1} - \sqrt{x+2}\):

\[ \frac{4}{\sqrt{x-1} + \sqrt{x+2}} \cdot \frac{\sqrt{x-1} - \sqrt{x+2}}{\sqrt{x-1} - \sqrt{x+2}} \] \[ = \frac{4(\sqrt{x-1} - \sqrt{x+2})}{(\sqrt{x-1})^2 - (\sqrt{x+2})^2} = \frac{4(\sqrt{x-1} - \sqrt{x+2})}{(x-1) - (x+2)} \] \[ = \frac{4(\sqrt{x-1} - \sqrt{x+2})}{x - 1 - x - 2} = \frac{4(\sqrt{x-1} - \sqrt{x+2})}{-3} = -\frac{4(\sqrt{x-1} - \sqrt{x+2})}{3} \]

Key Takeaways

When simplifying radical expressions, always look for perfect squares that can be factored out from under the radical. When radicals appear in denominators, use the conjugate technique to rationalize. Remember that the conjugate method works because multiplying conjugates produces a difference of squares, which eliminates the radical terms.