1. If two or more radicals are multiplied with the same index, you can take the radical once and multiply the numbers inside the radicals.
^{n}√a x ^{n}√b = ^{n}√(a x b)
2. If two radicals are in division with the same index, you can take the radical once and divide the numbers inside the radicals.
^{n}√a/^{n}√b = ^{n}√(a/b)
3. One number can be taken out of a square root for every two same numbers multiplied inside the square root. And also, one number can be taken out of a cube root for every three same numbers multiplied inside the cube root and so on.
√4 = √(2 x 2) = 2
^{3}√8 = ^{3}√(2 x 2 x 2) = 2
4. A radical with index n can be written as exponent 1/n.
^{n}√a = a^{1/n}
√a = a^{1/2}
^{3}√a = a^{1/3}
5. Addition and subtraction of two or more radicals can be performed with like radicals and like radicands only.
Like radicals - Radicals with the same index
Radicand - The number inside the radical
For example, 9√3 and 4√3 can be added or subtracted. Because the numbers inside the square roots are same.
9√3 + 4√3 = 13√3
9√3 - 4√3 = 5√3
6. If a radical with index n is moved from one side of the equation to the other side, it will become the exponent n.
^{n}√x = a
x = a^{n}
7. If an exponent n is moved from one side of the equation to the other side, it will become a radical with index n.
y^{n} = b
y = ^{n}√b
8. If the digit in one's place of a number is 2, 3, 7 or 8, then the number can not be a perfect square. So the square root of such numbers will be irrational.
For example, √23 = 4.795831.........
9. If a number ends with odd number of zeros, then, the square root of the number will be irrational.
For example, √3000 = 54.772255.......
10. The square root of a perfect square is always a rational number.
√4 = √(2 x 2) = 2
√25 = √(5 x 5) = 5
11. The square root of an even perfect square number is always even and the square root of an odd perfect square number is always is odd.
For example,
√64 = 8
√81 = 9
12. Square root of a negative number is considered to be an imaginary value.
For example, √(-2), √(-9).
Problem 1 :
Simplify :
√6 ⋅ √15
Solution :
= √6 ⋅ √15
= √(6 ⋅ 15)
= √(2 ⋅ 3 ⋅ 3 ⋅ 5)
= 3√(2 ⋅ 5)
= 3√10
Problem 2 :
Simplify :
√35 ÷ √7
Solution :
= √35 ÷ √7
= √(35/7)
= √5
Problem 3 :
Simplify :
3√425 + 4√68
Solution :
Decompose 425 and 68 into prime factors using synthetic division.
√425 = √(5 ⋅ 5 ⋅ 17) √425 = 5√17 |
√68 = √(2 ⋅ 2 ⋅ 17) √68 = 2√17 |
3√425 + 4√68 :
= 3(5√17) + 4(2√17)
= 15√17 + 8√17
= 23√17
Problem 4 :
Simplify :
√243 - 5√12 + √27
Solution :
Decompose 243, 12 and 27 into prime factors using synthetic division.
√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3
√12 = √(2 ⋅ 2 ⋅ 3) = 2√3
√27 = √(3 ⋅ 3 ⋅ 3) = 3√3
√243 - 5√12 + √27 :
= 9√3 - 5(2√3) + 3√3
= 9√3 - 10√3 + 3√3
= 2√3
Problem 5 :
Simplify :
√4 + ^{3}√27 + ^{4}√64
Solution :
√4 = √(2 ⋅ 2) = 2
^{3}√27 = ^{3}√(3 ⋅ 3 ⋅ 3) = 3
^{4}√625 = ^{4}√(5 ⋅ 5 ⋅ 5 ⋅ 5) = 5
√4 + ^{3}√27 + + ^{4}√64 :
= 2 + 3 + 5
= 10
Problem 6 :
Simplify :
^{3}√4 ⋅ ^{3}√16
Solution :
= ^{3}√4 ⋅ ^{3}√16
= ^{3}√(4 ⋅ 16)
= ^{3}√(4 ⋅ 4 ⋅ 4)
= 4
Problem 7 :
If ^{3}√a = 1/2, then find the value of a.
Solution :
^{3}√a = 1/2
a = (1/2)^{3}
a = 1^{3}/2^{3}
a = 1/8
Problem 8 :
If (^{3}√8)^{7} ⋅ (√2)^{-4} = 2^{k}, then solve for k.
Solution :
(^{3}√8)^{7} ⋅ (√2)^{-4} = 2^{k}
2^{7} ⋅ (2^{1/2})^{-4} = 2^{k}
2^{7} ⋅ 2^{-2} = 2^{k}
2^{7 - 2} = 2^{k}
2^{5} = 2^{k}
k = 5
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