How to Simplify Square Roots with Variables: A Step-by-Step Guide

Simplifying square roots, especially when variables are involved, can seem challenging at first. However, with the right approach and understanding of prime factorization, you can break down complex expressions into manageable parts. Let's go through the example of simplifying $$\sqrt{162x^2y^5}$$ step by step.

First, we begin with the number under the square root. The number 162 can be factored into its prime components. Here's the breakdown:

- Start with 162 and divide by 2: $$162 = 2 \times 81$$
- Then factor 81: $$81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3$$

This means we can express 162 as $$2 \times 3^4$$. Now we can simplify the square root of 162:

$$\sqrt{162} = \sqrt{2 \times 3^4}$$. Since $$\sqrt{3^4}$$ equals 9, we have:

$$\sqrt{162} = 9\sqrt{2}$$.

Next, we look at the variables in our expression. We have $$x^2$$ and $$y^5$$:
- The square root of $$x^2$$ is simply $$x$$, since the square root operation pulls out pairs of identical factors.
- For $$y^5$$, we can rewrite it as $$y^4 \cdot y$$. The square root of $$y^4$$ is $$y^2$$, so we can pull that out, leaving $$y$$ inside the square root.

Now, we can put everything together. From our simplifications:
- From $$\sqrt{162}$$ we get 9.
- From $$x^2$$ we get $$x$$.
- From $$y^5$$ we get $$y^2$$ outside and $$y$$ inside.

Combining all of these gives us:

$$9xy^2\sqrt{2y}$$.

However, we often express this in a more simplified format, resulting in:

$$3xy^2\sqrt{2y}$$.

This process not only helps in understanding the simplification of square roots but also illustrates the importance of understanding prime factorization and the properties of square roots. In real-world applications, simplifying square roots can be useful in fields such as engineering, physics, and finance, where calculations often involve square roots of numbers and variables.

Remember, practice makes perfect! Try simplifying other square roots with different variables and numbers to solidify your understanding. You'll soon find that simplifying square roots becomes a quick and easy task.