RULES FOR RADICALS
1. Multiplication
ⁿ√a × ⁿ√b = ⁿ√ab
You just have to multiply the numbers within the radical sign.
Ex.
²√8 × ²√2 = ²√16 = 4 ✔️
2. Division
ⁿ√a/b = ⁿ√a÷ⁿ√b
Ex. 1:
²√1/4 = ²√1 / ²√4 = 1/2 ✔️
Ex. 2:
³√8/4 = ³√8 / ³√4 = 2 / ³√4 ✔️
³√4 can no longer be simplified, since 4 has no perfect cube root.
3. Subtraction
Ex. 1:
2√3 - 6√3 = -4√3 ✔️
In subtracting, consider the numbers in the radical sign as the SURNAME/APELYIDO of the numbers. If their surnames are the same, then you can substract. Kung hindi magkadugo, magka-apelyido, wag mong agawan/bawasan.
Ex. 2:
√3 - √5 - √6 - 5√3
=-4√3 - √5 - √6 ✔️
That is the final answer since we can only subtract √3 and 5√3 since they have the same surnames, √3. But, we can no longer simplify the problem after which, since √3, √5, and √6 are different surnames already.
4. Addition
Ex. 1:
4√5 + 6√5 = 10√5 ✔️
Again, in adding radicals, the rules with subtraction will also apply that you can only add when their surnames are the same. If magkadugo, pwede mong dagdagan. Pwede mo ring agawan. Gaya ng kapatid mo, minsan inagawan mo ng pagkain, minsan dinagdagan mo naman. Treat the numbers in the radical signs as their surnames.
Ex. 2: Little Complicated
√2 + √8 = ?
In solving this, please observe the numbers inside the radicals if they are still factorable. √2 is no longer factorable, but √8 can still be factored by √4×2, thus,
√2 + √8 = ?
=√2 + √4×2 or √2 + √4 √2
=√2 + 2√2
=3√2 ✔️
5. Multiply Messy Radicals
Ex. 1:
(7-√2) (5+√3)
In solving this, you can use the First, Outside, Inside, Last (FOIL) Method.
F: 7×5 = 35
O: 7×√3 = 7√3
I: -√2 × 5 = -5√2
L: -√2 × √3 = -√6
Then, add afterwards:
=35 + 7√3 + (-5√2) + (-√6)
=35-5√2+7√3-√6 ✔️
This is the final answer since we can no longer simplify them since they have different surnames already, √2, √3, √6, and 35 even has no surname.
6. Rationalizing the Denominator of Radicals
How can you rationalize or remove the radical sign in the denominator, for example:
Ex. 1:
3/√2
This is the process.
Multiply a radical number in fraction which is still equal to 1 to the given number. In removing, the radical sign in √2, I should multiply it with a number that will result to a number with a perfect square, thus,
3/√2=?
=3/√2 × (√2/√2)
=3√2 / √4
=3√2 / 2 ✔️
Notice that we multiply √2/√2 to the number so that the denominator, √2, will become √4 which has a perfect square which is 2. Multiplying the number with √2/√2 does not distort its value and meaning since √2/√2 is still equal to 1. For example, 100 is the same with 100 * (2/2) since 2/2 is still equal to 1.
7. Rationalizing Denominator (Complex)
Ex. 1:
=[5/(2-√3)]
=[5/(2-√3)] × (2+√3/2+√3)
=5(2+√3)/(2-√3) (2+√3)
=10+5√3/ (4+2√3-2√3-√9)
=10+5√3/ (4-3)
=10+5√3 / 1
=10+5√3✔️
If the problem will look like with this example, you have to multiply the number with a number still equal to 1 but with DIFFERENT sign to the denominator. In our example, the denominator is 2-√3 but I multiplied it with 2+√3. That is the ruling.
8. √x² = x, only when x is equal or greater than 0
9. You can't find the square root of a negative number applying the rule in No. 3.
Ex. ²√-16 = undefined
10. You can find a cube root of a negative number.
Ex. 1:
=³√-8
=-2 ✔️
Since, -2 × -2 × -2 = -8
11. ⁿ√a × ⁿ√b = a^(1/n) b^(1/n) = (ab)^1/n = ⁿ√ab
12. (ⁿ√a)^m = ⁿ√a^m = a^m/n
m can be inside or outside radical signs.
Cipriano Vasig Romeral
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