GCSE Maths - What on Earth are Surds??? And How do You Simplify Them? (Part 1/3)

Summary

This video tutorial explains how to simplify surds, focusing on identifying and working with square numbers. It covers how to manipulate surds by using multiplication and division rules, as well as breaking down surds into factor pairs that include square numbers. By simplifying terms like √40, √27, and √200, viewers learn to transform complex surds into simpler expressions. The video emphasizes the importance of knowing square numbers and provides examples to make the process clearer and more approachable for learners.

Takeaways

• 😀 A third is an irrational root of a rational number, meaning it has a square root sign with a number inside that can't easily be squared.
• 😀 Square numbers, like 9 and 36, are not considered thirds because their square roots are whole numbers (3 and 6 respectively).
• 😀 Memorizing all square numbers is crucial for simplifying thirds efficiently.
• 😀 When multiplying square roots, you can combine them into a single square root (e.g., √3 * √5 = √15).
• 😀 You can also divide square roots by splitting them into smaller square roots (e.g., √40 = √4 * √10).
• 😀 The rule for combining or splitting square roots only works for multiplication and division, not for addition or subtraction.
• 😀 When simplifying square roots, factor them into pairs and look for a square number to simplify further.
• 😀 For example, √40 can be simplified by recognizing √4, turning it into 2√10.
• 😀 For √27, the factor pair √3 and √9 can be used, simplifying it to 3√3.
• 😀 For √200, the largest square factor (100) simplifies it to 10√2.
• 😀 The approach for √48 is to use √16, simplifying it to 4√3, making the process faster and easier.

Q & A

What is a third in mathematical terms?

A third is an irrational root of a rational number, which means it involves a square root sign with a number inside that is not a square number.

Why isn't the square root of 9 considered a third?

The square root of 9 is not considered a third because 9 is a square number, and its square root is 3, which is a rational number.

What are square numbers and why are they important for simplifying thirds?

Square numbers are numbers that can be written as the square of an integer (e.g., 1, 4, 9, 16, etc.). They are important for simplifying thirds because recognizing square factors allows us to simplify the root.

How does multiplication of thirds work?

When multiplying two thirds, you multiply the numbers inside the square roots. For example, √3 × √5 is equivalent to √(3 × 5) = √15.

Can thirds be divided like regular numbers?

Yes, thirds can be divided by using the same principle as multiplication. You can divide the square roots, such as √40 ÷ √2, which simplifies to √(40 ÷ 2) = √20.

What is the rule about adding or subtracting thirds?

You cannot add or subtract thirds directly by just combining the numbers under the square root. The rule only applies when you're multiplying or dividing thirds, not when adding or subtracting them.

Why is simplifying √40 into 2√10 helpful?

Simplifying √40 into 2√10 is helpful because 4 is a square number, and simplifying the square root of 4 into 2 makes the expression more manageable and easier to work with.

How do you simplify √27?

To simplify √27, you find its factor pair that contains a square number. Since 27 = 3 × 9 and 9 is a square number, you simplify √9 to 3, resulting in 3√3.

What is the biggest square factor when simplifying √200?

The biggest square factor of 200 is 100, because 100 is a square number and the largest factor of 200 that can be simplified. The square root of 100 is 10, so √200 simplifies to 10√2.

How do you simplify √48 most easily?

The easiest way to simplify √48 is to recognize that it can be written as √16 × √3, and since √16 is 4, the simplified form is 4√3.