Summary
Highlights
When adding or subtracting roots, only the numbers in front of identical roots are combined. The root itself remains unchanged. For example, 3√3 + 4√3 = 7√3. If there is no number before the root, it implies a '1', e.g., √2 is 1√2. This rule applies similarly to subtraction, where terms like 7√5 - √5 become 6√5.
For multiplication, numbers outside the root are multiplied by numbers outside, and numbers inside the root are multiplied by numbers inside. For example, 2√5 * 3√2 = (2*3)√(5*2) = 6√10. If roots are identical, like √2 * √2, the result is simply the number inside the root (in this case, 2), as √4 = 2. This method can simplify calculations.
Division of roots follows similar rules to multiplication: numbers outside the root are divided, and numbers inside the root are divided. For example, 4√6 / 2√2 = (4/2)√(6/2) = 2√3. If dividing by a root without a number in front, implicitly divide by 1. For fractions with roots, operations can be performed directly on the numbers and roots, simplifying where possible.
Exponetting roots means multiplying the root by itself the number of times indicated by the exponent. For example, (2√3)² = (2√3) * (2√3). This results in 2*2 * √3*√3 = 4 * 3 = 12. If a root is raised to a power, such as √2³, it can be written as √(2*2*2) = √8, which can then be simplified to 2√2.
To simplify a root, find factors of the number under the root, where at least one factor is a perfect square. For √8, it can be written as √(4*2) = √4 * √2 = 2√2. An alternative method is prime factorization: divide the number by prime numbers until you reach 1. For every pair of identical prime factors, one factor is extracted from under the root. Any leftover prime factors remain under the root.
To remove a root from the denominator, multiply both the numerator and the denominator by the root present in the denominator. For example, 2/√3 is multiplied by √3/√3, resulting in (2√3)/(√3*√3) = 2√3/3. This method ensures that no root remains in the denominator. If the numerator is a binomial, a bracket must be used for multiplication.
The video concludes by solving example problems typical for the 8th-grade exam. These involve combining multiple operations covered, such as multiplication and subtraction, and then comparing the final values. Memorizing a 'root multiplication table' (like standard multiplication tables for numbers) can save time and reduce errors in exams.