What is Rationalising the Denominator? | Simple Explanation for Class 6

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What is Rationalising the Denominator of a Monomial Surd?

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Rationalising the denominator of a monomial surd means removing the square root (or any root) from the bottom part (denominator) of a fraction. We do this to make calculations easier and to present the answer in a standard, simpler form, without a surd in the denominator.

Simple Example

Quick Example

Imagine you have a fraction like 1/sqrt(2) representing a part of your mobile data usage. It's tricky to divide 1 by 'square root of 2' directly. Rationalising it means changing it to a form like sqrt(2)/2, which is much easier to understand and use in calculations, just like converting rupees to paise makes it easier to count small amounts.

Worked Example

Step-by-Step

Let's rationalise the denominator of 3/sqrt(5).

Step 1: Identify the surd in the denominator. Here it is sqrt(5).

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Step 2: Multiply both the numerator (top) and the denominator (bottom) by this surd. Remember, multiplying by sqrt(5)/sqrt(5) is like multiplying by 1, so the value of the fraction doesn't change.
(3 / sqrt(5)) * (sqrt(5) / sqrt(5))

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Step 3: Multiply the numerators together: 3 * sqrt(5) = 3sqrt(5).

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Step 4: Multiply the denominators together: sqrt(5) * sqrt(5) = 5.

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Step 5: Write the new fraction with the calculated numerator and denominator.
3sqrt(5) / 5

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Answer: The rationalised form of 3/sqrt(5) is 3sqrt(5)/5.

Why It Matters

Rationalising helps simplify complex calculations in fields like Physics when dealing with waves or light. Engineers use it to design efficient structures, and even in Computer Science, it helps in writing cleaner code for calculations. Learning this makes your math foundation strong for future studies and exciting careers!

Common Mistakes

MISTAKE: Only multiplying the denominator by the surd. For example, changing 1/sqrt(3) to 1/3. | CORRECTION: You must multiply BOTH the numerator and the denominator by the surd to keep the fraction's value the same. So, 1/sqrt(3) becomes (1 * sqrt(3)) / (sqrt(3) * sqrt(3)) = sqrt(3)/3.

MISTAKE: Thinking sqrt(a) * sqrt(a) = sqrt(a). For example, sqrt(7) * sqrt(7) = sqrt(7). | CORRECTION: Remember that sqrt(a) * sqrt(a) = a. So, sqrt(7) * sqrt(7) = 7.

MISTAKE: Forgetting to simplify the final fraction if possible. For example, leaving 6sqrt(2)/2 as the answer. | CORRECTION: Always check if the whole number part of the numerator and the denominator can be divided. 6sqrt(2)/2 simplifies to 3sqrt(2).

Practice Questions

Try It Yourself

QUESTION: Rationalise the denominator of 2/sqrt(7). | ANSWER: 2sqrt(7)/7

QUESTION: Rationalise the denominator of 10/sqrt(2). | ANSWER: 5sqrt(2)

QUESTION: Rationalise the denominator of 4/(3sqrt(5)). | ANSWER: 4sqrt(5)/15

MCQ

Quick Quiz

Which of the following is the rationalised form of 5/sqrt(10)?

5sqrt(10)

sqrt(10)/2

50/sqrt(10)

1/sqrt(2)

The Correct Answer Is:

To rationalise 5/sqrt(10), multiply both numerator and denominator by sqrt(10). This gives (5 * sqrt(10)) / (sqrt(10) * sqrt(10)) = 5sqrt(10)/10, which simplifies to sqrt(10)/2.

Real World Connection

In the Real World

When calculating the precise length of cables for a new metro line or the dimensions for a drone delivery route in a city like Bangalore, engineers often encounter expressions with surds. Rationalising these helps them get exact, easy-to-use measurements for construction and programming, ensuring everything fits perfectly.

Key Vocabulary

Key Terms

SURD: A number that cannot be expressed exactly as a simple fraction or integer, usually involving a root (like sqrt(2)). | DENOMINATOR: The bottom number in a fraction. | NUMERATOR: The top number in a fraction. | RATIONALISE: To remove surds from the denominator of a fraction. | MONOMIAL: An algebraic expression consisting of only one term.

What's Next

What to Learn Next

Great job learning about rationalising monomial surds! Next, you can explore 'Rationalising the Denominator of a Binomial Surd'. This builds on what you've learned and helps you tackle even more complex fractions!