Introduction
Simplifying square roots can often be a daunting task for students of all levels. With numerous steps and complex calculations, it’s no wonder that some individuals find it challenging to master. However, with a little guidance and practice, simplifying square roots can become second nature. In this article, we will cover five simple steps to simplifying a square root, some tricks to make the process easier and quicker, a comprehensive guide to mastering the art of simplifying square roots, how to tackle more complex problems, and the top mistakes to avoid. Let’s dive in!
5 Simple Steps to Simplify a Square Root
Before we dive into the more complex and advanced techniques, let’s go over the five simple steps needed to simplify a square root:
Step 1: Identify the Perfect Square Factors
The first step to simplifying a square root is identifying the perfect square factors. These are numbers whose square roots are whole numbers. Examples of perfect square factors include 4, 9, 16, 25, and so on.
For instance, you can identify the perfect square factors of 72 by listing the first few perfect square numbers: 1, 4, 9, 16, and so on, then identifying which perfect squares divide into the original number with no remainders. We find that both 4 and 9 divide into the original number: $$72 = 4\times18$$.
Step 2: Rewrite the Square Root as the Product of Perfect Square Factors
Once you have identified the perfect square factors of the original number, you can now rewrite the square root as a product of these factors. You should also include any other remaining non-perfect square factors.
For example, if our original number is 72, we can rewrite it as the product of perfect square factors and remaining factors as follows: $$\sqrt{72} = \sqrt{4\times18} = \sqrt{4}\times\sqrt{18}$$. Note that 4 is a perfect square factor, and 18 is not a perfect square factor.
Step 3: Simplify the Perfect Square Factors Outside of the Square Root
You can now simplify the perfect square factors outside of the square root, using the known values of their square roots.
Inserting the square root of 4 from the previous example, we get: $$\sqrt{72} = 2\sqrt{18}$$. This expression is much simpler than the initial square root of 72, but it’s not quite there yet.
Step 4: Combine Any Remaining Non-Perfect Square Factors Inside the Square Root
Next, combine any remaining non-perfect square factors inside the square root by multiplying them together.
For our example of 72, we can combine the remaining non-perfect square factor of 18 inside the square root: $$\sqrt{72} = 2\sqrt{18} = 2\sqrt{2\times9} = 2\sqrt{2}\times\sqrt{9} = 6\sqrt{2}.$$ The expression has been simplified down to a single term, which can be useful in resizing or performing additional calculations.
Step 5: Check Your Answer
The final step is to check your answer by squaring it and comparing it to the original number.
For our example of 72 do: $$6\sqrt{2}\simeq 12.73$$
After using your chosen method to perform the square operation, the result should match the original number. If the two do not match, there may have been a mistake in the initial calculation.
The Tricks to Simplifying Square Roots You Didn’t Learn in School
While the five steps mentioned above are essential to simplifying square roots, there are also several tricks and shortcuts that can make the process even more straightforward.
Trick 1: Identify the Perfect Square Factors Based on their Last Digits
One of the quickest ways to identify perfect square factors is by looking at the last digit of the original number. For example, if the last digit of the number is 1, 4, 5, 6, or 9, it is a perfect square.
Let’s consider our example of 72 with this trick. Since the last digit of 72 is 2, it may not seem like there are any perfect square factors that will cleanly divide into the number. But if we look closer, we will discover that 72 is actually 9 less than a multiple of 25, or $$5^2$$. Therefore, we can break up the number into the perfect square of 25 minus the non-square leftover, 9: $$72 = 5^2\times1 + 9$$.
Once you identify that the 9 is the remaining factor after removing $$5^2$$ you can simplify: $$\sqrt{72} = \sqrt{5^2\times1+9} = 5\sqrt{1}+\sqrt{9} = 5+3 = 8 \text{ .} $$
Trick 2: Break Down the Square Root
Another helpful trick is to break down the square root into smaller, more manageable parts.
Let’s say we had to simplify $$\sqrt{1008}$$. An excellent way to begin simplification would be to examine the factors of 1008 using an appropriate technique. By performing prime factorization, we determine that $$1008 = 2^4\times3^2\times7$$. Now let’s apply the first simple step in the simplification of square roots by identifying perfect square factors. We find that $$2^2 = 4 \text{;}\; 3^2 = 9$$. Thus, we can rewrite our initial problem as:
$$\sqrt{1008} = \sqrt{4\times9\times7\times4\times4\times4} = \sqrt{4^3\times3^2\times7}.$$
We can now extract the perfect square factors and simplify:
$$\sqrt{1008} = 4\sqrt3\sqrt7$$
Mastering the Art of Simplifying Square Roots: A Comprehensive Guide
Now that we have covered the basics and provided some tricks, let’s take a deep dive into the art of simplifying square roots. This section breaks down the fundamental concepts and more complex techniques that you need to know to simplify square roots with ease.
Start with Simplification Techniques for Non-Complex Square Roots
We must first understand the fundamental concepts of simplifying square roots before moving on to more advanced techniques. Several strategies can quickly simplify square roots; or allow you to recognize when a root is already simplified.
On the other hand, supplementary algebraic techniques, such as simplification by factoring, are necessary for more complicated square roots.
Algebraic Techniques for Complex Square Roots
When it comes to more complex square roots, the simplification process is often more involved and requires additional algebraic techniques.
One of the most effective methods is to simplify using factoring, which involves breaking down the number inside the square root into its components and then factoring each component as far as possible.
Square Roots Simplified: How to Tackle Tricky Problems with Ease
Now that you have covered the basics of simplifying square roots and some advanced techniques let us tackle more complex problems
Fractional Square Roots
Suppose we had to simplify $$\sqrt{16\div25}$$. We can solve this problem by simplifying the fraction first: $$\sqrt{16\div25} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$$
Variable Square Roots
Suppose we needed to simplify $$\sqrt{12x^2}$$
We can first break down the square root by finding the factors of the number inside. In this case, the factors are 4 and 3, so we can write:
$$\sqrt{12x^2} = \sqrt{4\times3\times x^2} = \sqrt{4}\times\sqrt{3} \times \sqrt{x^2} = 2\sqrt3\text{ }\left|x\right|$$
Note that we include the absolute value signs when dealing with variable square roots to account for negative values of x.
The Top Mistakes to Avoid When Simplifying Square Roots
As with any mathematical concept, there are common mistakes that individuals make when simplifying square roots. By being aware of these mistakes, you can improve your accuracy and efficiency in simplifying square roots.
Mistake 1: Failing to Simplify Perfect Square Factors
One of the biggest mistakes individuals make when simplifying square roots is failing to simplify perfect square factors outside of the square root. It is essential to simplify any perfect square numbers or factors before bringing them outside of the square root sign.
Mistake 2: Forgetting the Absolute Value for Variable Square Roots
As we have seen in the more complex problems section, always include the absolute value for variable square roots when there is the possibility of negative values of x.
Mistake 3: Combining Two Perfect Squares Incorrectly
Another common mistake when simplifying square roots is improperly combining two perfect squares. Many individuals will simply add or subtract the numbers, but this is incorrect. When two perfect squares are being added or subtracted inside a square root, both numbers must be multiplied by each other.
Conclusion
By using the five simple steps and some useful tricks, individuals can easily simplify square roots. By mastering the five steps and avoiding common mistakes, you can develop the necessary skills to tackle more complex square-root related problems with ease. Remember that practice is key in mastering this topic, so always look for additional opportunities to exercise your newfound knowledge.
