Welcome back, class! The past week, we have been deep in the weeds looking at solving quadratic equations using factoring. As you recall, factoring is a fantastic method when you have more than one variable term in the problem (for example, an $x^2$ term and an $x$ term), because you cannot simply isolate the variable.

However, when you only have one variable term in the problem (just an $x^2$), the square root method is often much faster and more efficient. Today, we are setting the foundation for this method by learning how to simplify square roots properly.

Why Simplify?

Just like we reduce fractions to their simplest form (e.g., $\frac{4}{8}$ becomes $\frac{1}{2}$), we must also simplify radicals. A square root is considered simplified when there are no perfect square factors left under the radical symbol.

Recall your perfect squares:

  • $2^2 = 4$
  • $3^2 = 9$
  • $4^2 = 16$
  • $5^2 = 25$
  • $6^2 = 36$
  • ...and so on.

How to Simplify Square Roots

To simplify a square root, look for the largest perfect square that divides evenly into the number under the radical. Let's look at a few examples from your homework worksheet.

Example 1: $\sqrt{45}$ (Problem #7)

First, ask yourself: Which perfect square goes into 45? Looking at our list, we know that $9$ goes into $45$.

$$ \sqrt{45} = \sqrt{9 \cdot 5} $$

Since the square root of 9 is 3, we can bring that outside the radical:

$$ = 3\sqrt{5} $$

Example 2: $\sqrt{96}$ (Problem #1)

This one is a bit trickier. You might notice that 4 goes into 96, but is there a larger one? Yes, 16 goes into 96.

$$ \sqrt{96} = \sqrt{16 \cdot 6} $$

Because $\sqrt{16} = 4$, the simplified form is:

$$ = 4\sqrt{6} $$

Example 3: Dealing with Coefficients like $10\sqrt{96}$ (Problem #11)

If there is already a number outside the radical, simply multiply it by whatever you bring out. We already know from the previous example that $\sqrt{96} = 4\sqrt{6}$.

$$ 10\sqrt{96} = 10 \cdot (4\sqrt{6}) = 40\sqrt{6} $$

Discussion Question: Domain Restrictions

Square roots are unique because they have a strict domain restriction. As we discussed in the first cycle regarding functions, you have to be careful about what you plug into a square root.

Question for the forums: What happens if you try to take the square root of a negative number in the real number system? Find a website or video that explains what happens when you plug a value that is not in the domain into a square root function, and share your findings.

Homework Assignment