How To Solve: Roots

Attached pdf of this Article as SPOILER at the top! Happy learning!

Hi All,

I have recently uploaded a video on YouTube to discuss Roots in Detail:




Following is covered in the video

¤ Square root of a number is ALWAYS positive
¤ Simplifying √a * √b
¤ Simplifying \(\frac{√a}{√b}\)
¤ Simplifying \((√a)^n \)
¤ Simplifying √(x ± y)
¤ Simplifying \(a^{(x/y)}\)
¤ Simplifying \(√(a^2)\)
¤ Simplifying √n

Square root of a number is ALWAYS positive

Even root of any number will always be a positive value
Ex: √36 = + 6

Although, \(x^2\) = 36 => x = ± √36 = ±6

Simplifying √a * √b

√a * √b = √(ab)
Ex: √2 * √3 = √(2*3) = √6

Simplifying \(\frac{√a}{√b}\)

\(\frac{√a}{√b}\) = \(√(\frac{a}{b})\)
Ex: \(\frac{√4}{√2}\) = \(√(\frac{4}{2})\) = √2

Simplifying \((√a)^n \)

\((√a)^n\) = √(\(a^n\))
Ex: \((√2)^4\) = √(\(2^4\)) = \(2^2\) = 4

Simplifying √(x ± y)

√(x ± y) ≠ √x ± √y
Ex: √(2 + 3) ≠ √2 + √3
=> √5 ≠ √2 + √3
=> 2.23 ≠ 1.414 + 1.732
=> 2.23 ≠ 3.146

Simplifying \(a^{(x/y)}\)

\(a^{(x/y)}\) = \(\sqrt[y]{a^x}\)
Ex: \(2^{(6/3)}\) = \(\sqrt[3]{2^6}\) = \(2^2\) = 4

Simplifying \(√(a^2)\)

√(\(a^2\)) = |a|
√(\(a^2\)) = -a, when a ≤ 0
√(\(a^2\)) = a, when a ≥ 0

Ex: √(\(3^2\)) = +3 = |3|
√(\((-3)^2\)) = 3 = -(-3) = |-3|

Simplifying √n

To simplify √n, we need to express n in powers of prime numbers and then need to take out the even powers.

Example: √56 = √(4*14) = √((2^2) * 14) = 2√14

Hope it helps!
Good Luck!