Rationalize The Denominator: Examples & Solutions

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Rationalize the Denominator: Examples & Solutions

Hey guys! Today, let's dive into the awesome world of rationalizing denominators. It's a crucial skill in mathematics that simplifies expressions and makes them easier to work with. Basically, it means getting rid of any pesky square roots (or other radicals) from the bottom of a fraction. Sounds cool, right? So, let's jump right in and break it down with some examples. We will be rationalizing two expressions today which are 852\frac{8}{5 \sqrt{2}} and 125−1\frac{12}{\sqrt{5}-1}.

Rationalizing Single-Term Denominators

Okay, so first, let's tackle fractions where the denominator has just one term, like our first example: 852\frac{8}{5 \sqrt{2}}. The main goal here is to eliminate the square root from the denominator. We can do this by multiplying both the numerator and the denominator by the square root that's causing the problem. In this case, that's 2\sqrt{2}. Remember, multiplying the top and bottom of a fraction by the same value doesn't change the fraction itself – it's like multiplying by 1.

So, here’s how it works:

852∗22=825∗2=8210\frac{8}{5 \sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{5 * 2} = \frac{8\sqrt{2}}{10}

Now, we can simplify the fraction 8210\frac{8\sqrt{2}}{10} by dividing both the numerator and the denominator by their greatest common factor, which is 2. This gives us:

8210=425\frac{8\sqrt{2}}{10} = \frac{4\sqrt{2}}{5}

And that’s it! We've successfully rationalized the denominator. The final expression is 425\frac{4\sqrt{2}}{5}, and there's no square root in the bottom. Easy peasy, right?

Why This Works

You might be wondering why multiplying by the square root works. When you multiply a square root by itself, you get rid of the root. For example, 2∗2=2\sqrt{2} * \sqrt{2} = 2. This is because 2\sqrt{2} is the number that, when multiplied by itself, equals 2. By multiplying the denominator by 2\sqrt{2}, we transform it into a rational number (a number without a square root).

Key Steps to Remember

  1. Identify the Radical: Spot the square root (or other radical) in the denominator.
  2. Multiply: Multiply both the numerator and the denominator by that radical.
  3. Simplify: Simplify the resulting fraction, if possible, to its simplest form.

Rationalizing Two-Term Denominators

Now, let's move on to something a bit more challenging: rationalizing denominators that have two terms, like our second example: 125−1\frac{12}{\sqrt{5}-1}. This type of problem requires a slightly different approach. We use something called the conjugate. The conjugate of a binomial expression (an expression with two terms) is the same expression but with the opposite sign in the middle.

For 5−1\sqrt{5} - 1, the conjugate is 5+1\sqrt{5} + 1. We multiply both the numerator and the denominator by this conjugate. This might seem a bit weird, but it helps us get rid of the square root in the denominator using the difference of squares formula: (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2.

Here’s how it looks:

125−1∗5+15+1=12(5+1)(5−1)(5+1)\frac{12}{\sqrt{5}-1} * \frac{\sqrt{5}+1}{\sqrt{5}+1} = \frac{12(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)}

Now, let's multiply out the denominator:

(5−1)(5+1)=(5)2−(1)2=5−1=4(\sqrt{5} - 1)(\sqrt{5} + 1) = (\sqrt{5})^2 - (1)^2 = 5 - 1 = 4

So, our expression becomes:

12(5+1)4\frac{12(\sqrt{5}+1)}{4}

We can simplify this by dividing both the numerator and the denominator by their greatest common factor, which is 4:

12(5+1)4=3(5+1)\frac{12(\sqrt{5}+1)}{4} = 3(\sqrt{5}+1)

Distribute the 3 and you get:

35+33\sqrt{5} + 3

And there you have it! We've successfully rationalized the denominator of 125−1\frac{12}{\sqrt{5}-1}. The result is 35+33\sqrt{5} + 3, with no square roots in the denominator.

Why the Conjugate Works

The conjugate works because it utilizes the difference of squares formula. When you multiply (a−b)(a - b) by (a+b)(a + b), you get a2−b2a^2 - b^2. This eliminates the square root because you're squaring it. In our example, (5−1)(5+1)(\sqrt{5} - 1)(\sqrt{5} + 1) becomes (5)2−12(\sqrt{5})^2 - 1^2, which simplifies to 5−1=45 - 1 = 4, a rational number.

Key Steps to Remember

  1. Identify the Conjugate: Find the conjugate of the denominator by changing the sign between the terms.
  2. Multiply: Multiply both the numerator and the denominator by the conjugate.
  3. Simplify: Simplify the resulting fraction, using the difference of squares formula, if applicable, and reduce the fraction to its simplest form.

Additional Tips and Tricks

  • Always Simplify: After rationalizing, always check if you can simplify the fraction further. Look for common factors between the numerator and the denominator.
  • Practice Makes Perfect: The more you practice, the easier it will become to recognize when and how to rationalize denominators. Try different examples and challenge yourself.
  • Watch Out for Signs: Pay close attention to the signs when finding the conjugate. It’s easy to make a mistake, but getting the sign right is crucial.
  • Complex Fractions: Sometimes, you might encounter fractions within fractions. Simplify these before attempting to rationalize the denominator.

Common Mistakes to Avoid

  • Forgetting to Multiply the Numerator: Always multiply both the numerator and the denominator by the same value. Multiplying only the denominator changes the value of the fraction.
  • Incorrectly Identifying the Conjugate: Make sure you change the correct sign when finding the conjugate. It's the sign between the terms that needs to be flipped.
  • Skipping Simplification: Don't forget to simplify the fraction after rationalizing. Sometimes, you can reduce the fraction to a simpler form.
  • Distributing Incorrectly: When multiplying the conjugate, be careful to distribute correctly, especially when there are multiple terms in the numerator or denominator.

Conclusion

So, there you have it! Rationalizing denominators might seem tricky at first, but with a little practice, it becomes second nature. Remember, the goal is to eliminate those pesky square roots (or other radicals) from the bottom of the fraction. Whether you're dealing with single-term or two-term denominators, the key is to multiply by the appropriate value (either the radical itself or the conjugate) and simplify. Keep practicing, and you'll be a pro in no time! Keep up the great work, guys, and happy calculating!