Simplifying Radicals A Step-by-Step Guide To √(98/200) And More
Hey guys! Today, we're diving into the exciting world of simplifying radicals. Radicals might seem intimidating at first, but trust me, with a few key steps, you'll be simplifying them like a pro. We’re going to break down how to simplify radicals, focusing on examples like √(98/200) and many others. Let’s get started!
What are Radicals?
Before we jump into the nitty-gritty, let's quickly recap what radicals are. In mathematics, a radical is a symbol (√) that indicates the root of a number. The most common radical is the square root, which asks, "What number multiplied by itself equals the number under the radical?" For example, √25 asks, "What number times itself equals 25?" The answer, of course, is 5.
Simplifying radicals means expressing them in their simplest form. This usually involves removing any perfect square factors from under the radical sign. Now, let’s tackle some examples!
a) Simplifying √(98/200)
Let's start with our first example: √(98/200). This might look a bit daunting, but don’t worry, we'll take it step by step.
Step 1: Simplify the Fraction Inside the Radical
Our first goal is to simplify the fraction inside the radical. Look for common factors in both the numerator (98) and the denominator (200). Both 98 and 200 are even numbers, so we can start by dividing both by 2:
98 ÷ 2 = 49
200 ÷ 2 = 100
So, our fraction simplifies to 49/100. Now, we have:
√(98/200) = √(49/100)
Step 2: Take the Square Root of the Numerator and Denominator
Now that we've simplified the fraction, we can take the square root of the numerator and the denominator separately. Ask yourself: what is the square root of 49, and what is the square root of 100?
The square root of 49 is 7, because 7 * 7 = 49.
The square root of 100 is 10, because 10 * 10 = 100.
So, we have:
√(49/100) = 7/10
That’s it! √(98/200) simplified is 7/10. See? Not so scary after all!
b) Simplifying √(216/882)
Next up, we have √(216/882). This one looks a bit trickier, but we'll use the same approach:
Step 1: Simplify the Fraction Inside the Radical
First, let's find common factors between 216 and 882. Both numbers are even, so we can divide by 2:
216 ÷ 2 = 108
882 ÷ 2 = 441
So, we have:
√(216/882) = √(108/441)
Now, we need to keep simplifying. Both 108 and 441 are divisible by 9:
108 ÷ 9 = 12
441 ÷ 9 = 49
Our fraction now looks like this:
√(108/441) = √(12/49)
Step 2: Take the Square Root of the Numerator and Denominator
We can take the square root of the denominator (49) easily, which is 7. However, 12 is not a perfect square. We need to simplify √12 first.
√12 can be written as √(4 * 3). The square root of 4 is 2, so we have:
√12 = 2√3
Now, we can rewrite our expression:
√(12/49) = (2√3) / 7
So, √(216/882) simplified is (2√3) / 7. Great job!
c) Simplifying √(225/196)
Let’s move on to √(225/196). This one is quite straightforward if you know your perfect squares.
Step 1: Simplify the Fraction Inside the Radical
In this case, 225 and 196 don't have any common factors other than 1, so the fraction is already in its simplest form.
Step 2: Take the Square Root of the Numerator and Denominator
Now, let’s take the square root of both the numerator and the denominator:
√225 = 15 (because 15 * 15 = 225)
√196 = 14 (because 14 * 14 = 196)
So, we have:
√(225/196) = 15/14
√(225/196) simplifies to 15/14. Easy peasy!
d) Simplifying √(144/720)
Now, let's tackle √(144/720). This one will give us a chance to practice our simplification skills.
Step 1: Simplify the Fraction Inside the Radical
First, we need to find common factors between 144 and 720. Both numbers are divisible by 144:
144 ÷ 144 = 1
720 ÷ 144 = 5
So, our fraction simplifies to 1/5:
√(144/720) = √(1/5)
Step 2: Take the Square Root of the Numerator and Denominator
The square root of 1 is 1. However, we have √5 in the denominator, which is not a perfect square. To rationalize the denominator, we'll multiply both the numerator and the denominator by √5:
√(1/5) = (1/√5) * (√5/√5) = √5 / 5
So, √(144/720) simplifies to √5 / 5.
e) Simplifying √(588/686)
Let's move on to √(588/686). This one looks a bit challenging, but we've got this!
Step 1: Simplify the Fraction Inside the Radical
First, let’s find common factors between 588 and 686. Both numbers are even, so we can start by dividing by 2:
588 ÷ 2 = 294
686 ÷ 2 = 343
So, we have:
√(588/686) = √(294/343)
Now, we need to find more common factors. Both 294 and 343 are divisible by 7:
294 ÷ 7 = 42
343 ÷ 7 = 49
Our fraction now looks like this:
√(294/343) = √(42/49)
We can simplify further. Both 42 and 49 are divisible by 7:
42 ÷ 7 = 6
49 ÷ 7 = 7
So, we have:
√(42/49) = √(6/7)
Step 2: Take the Square Root of the Numerator and Denominator
We have √(6/7). To rationalize the denominator, we'll multiply both the numerator and the denominator by √7:
√(6/7) = (√6/√7) * (√7/√7) = √(6*7) / 7 = √42 / 7
So, √(588/686) simplifies to √42 / 7.
f) Simplifying √(675/648)
Last but not least, we have √(675/648). Let's finish strong!
Step 1: Simplify the Fraction Inside the Radical
First, we need to find common factors between 675 and 648. Both numbers are divisible by 9:
675 ÷ 9 = 75
648 ÷ 9 = 72
So, we have:
√(675/648) = √(75/72)
Now, let’s simplify further. Both 75 and 72 are divisible by 3:
75 ÷ 3 = 25
72 ÷ 3 = 24
Our fraction now looks like this:
√(75/72) = √(25/24)
Step 2: Take the Square Root of the Numerator and Denominator
The square root of 25 is 5. However, 24 is not a perfect square. We need to simplify √24.
√24 can be written as √(4 * 6). The square root of 4 is 2, so we have:
√24 = 2√6
Now, we can rewrite our expression:
√(25/24) = 5 / (2√6)
To rationalize the denominator, we'll multiply both the numerator and the denominator by √6:
5 / (2√6) * (√6/√6) = 5√6 / (2*6) = 5√6 / 12
So, √(675/648) simplifies to 5√6 / 12.
Conclusion
And there you have it! We've successfully simplified radicals like √(98/200), √(216/882), √(225/196), √(144/720), √(588/686), and √(675/648). Remember, the key is to simplify the fraction inside the radical first and then look for perfect square factors. With practice, you'll become a radical-simplifying superstar!
Keep practicing, and you’ll master these skills in no time. You've got this!
