[Mathematics] 3rd Year: Square Root Master Guide
Hello, everyone! Today, we’re going to learn about a new concept called "square roots." You might have heard the term "root" before, right?
It might sound a bit intimidating at first, but it’s actually based on the simple idea of "finding the length of one side of a square from its area." If you take it one step at a time, you’re bound to become a pro! Let's do our best together.
1. What is a square root?
Think about the area of a square
Imagine a square with an area of 9. What is the length of one side?
The answer is 3 cm, because \(3 \times 3 = 9\) (3 squared is 9).
In this case, we say that 3 is a square root of 9.
Now, what about a square with an area of 5?
Since \(2 \times 2 = 4\) and \(3 \times 3 = 9\), the length must be a number between 2 and 3, but there isn't a perfect integer.
To solve this, we decided to use a new symbol called "\(\sqrt{}\)" (radical sign/root) and write it as \(\sqrt{5}\).
Definition of a square root
When a number \(x\) is squared to become \(a\), we call \(x\) the square root of \(a\).
Tip: Most positive numbers have two square roots—one positive and one negative!
Example: The square roots of 9 are 3 and -3. (This is because \((-3) \times (-3)\) also equals 9.)
We can write these together as \(\pm 3\).
【Common Mistake!】
The answer to "\(\sqrt{9}\)" is only 3!
If a question asks for "the square roots of 9," you must answer both 3 and -3. However, "\(\sqrt{9}\)" specifically refers only to the positive one. Be careful; this is a common trap on tests!
2. Comparing square roots
Comparing magnitudes
If the number inside the root is larger, the value itself is larger.
If \(a < b\), then \(\sqrt{a} < \sqrt{b}\).
Example: \(\sqrt{2} < \sqrt{5}\)
Pro-tip: How to compare a root to a non-root number
If you're asked, "Which is larger, 3 or \(\sqrt{7}\)?", just square both numbers!
・\(3^2 = 9\)
・\((\sqrt{7})^2 = 7\)
Since 9 is larger than 7, we know that \(3 > \sqrt{7}\). The trick is to remove the root and put them on equal footing!
3. Rational and Irrational Numbers
Numbers can be divided into two main groups.
① Rational Numbers
Numbers that can be expressed as a fraction (\(\frac{\text{integer}}{\text{integer}}\)). This includes integers and decimals.
Examples: \(2, -0.5, \frac{1}{3}, \sqrt{16}\) (since \(\sqrt{16} = 4\), it's a rational number)
② Irrational Numbers
Numbers that cannot be expressed as a fraction. If written as a decimal, the digits go on forever without a repeating pattern.
Examples: \(\pi\) (pi), \(\sqrt{2}, \sqrt{3}\), etc.
【Key Point】
Any number whose root cannot be simplified into an integer is an irrational number!
4. Calculation Rules for Square Roots
This is where the actual calculation begins. The rules are simple, so think of it like solving a puzzle.
① Multiplication and Division
For multiplication and division of roots, you can just calculate the numbers inside the root together!
・\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
・\(\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}\)
② Simplifying \(\sqrt{a^2b} = a\sqrt{b}\)
Any "pair of squares" inside the root can be moved outside.
Example: \(\sqrt{12} = \sqrt{2^2 \times 3} = 2\sqrt{3}\)
This process of "making the number inside the root as small as possible" is the absolute fundamental rule of calculations!
③ Rationalizing the Denominator
When there is a root in the denominator, multiply both the numerator and denominator by the same root to remove it from the bottom.
Example: \(\frac{1}{\sqrt{2}} = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}\)
This is called "rationalizing." Always rationalize your answers!
④ Addition and Subtraction
Watch out! \(\sqrt{2} + \sqrt{3} = \sqrt{5}\) is incorrect!
Addition and subtraction with roots follow the same rules as algebraic expressions (like \(2x + 3x = 5x\)).
You can only calculate terms if the numbers inside the roots are the same.
Example: \(2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\)
Example: \(\sqrt{2} + \sqrt{3}\) cannot be calculated any further.
5. Useful Approximate Values to Remember
Knowing roughly what these roots equal makes solving problems much easier. Here are some famous Japanese mnemonics!
・\(\sqrt{2} \approx 1.41421356\) (A standard mnemonic is "hitoyo hitoyo ni hitomigoro")
・\(\sqrt{3} \approx 1.7320508\) ("hito-nami ni ogore ya")
・\(\sqrt{5} \approx 2.2360679\) ("fujisan-roku oumu naku")
【Summary of Key Points】
1. A square root is a number that, when squared, gives you the original number (there are both positive and negative ones!).
2. You can only add or subtract roots when the numbers inside them are the same.
3. Always "simplify the inside of the root" and "rationalize the denominator" at the end of your calculations.
At first, it might take you a while to convert \(\sqrt{12}\) to \(2\sqrt{3}\). But as you solve more practice problems, your brain will get used to it naturally. Practice makes perfect! Don't rush, just take it one step at a time. I'm rooting for you!