ROOTS OF INTEGERS
About "Roots of integers"
Roots of integers :
To find roots of integers, first we have to notice the index of the given root. According to the index of the given radical, we have to split the number and simplify.
What is index of the root ?
We have to read the first term as cube root of 27 and second term as 5th root of 125.
How to find roots of integers
Step 1 :
Let "x" be the root of the given radical term
Step 2 :
According to the index, we have to raise power on both sides.For example, if we have square root, then we have to take squares on both sides, in order to remove the square root.
Step 3 :
After cancelling the power and the radical sign in right hand side, we have to express the number in the exponential form.
Step 4 :
Check whether we have same powers on either sides of equal sign.
Step 5 :
Since the power are equal, we can decide that bases are also equal.
From the above steps, we can find the roots of the integers.
Let us look into some examples based on the above concept.
Example 1 :
Find the real number root of √64
Solution :
Step 1 :
Index of the given square root = 2
Let "x" be the required root
x = √64
Step 2 :
In order to remove the square root, take squares on both sides
x² = (√64)²
x² = 64
Step 3 :
Express 64 as the product of two same numbers
That is, 64 = 8 x 8
64 = 8²
x² = 8²
Step 4 :
Since the powers are equal, then their base are also equal.
x = 8
Hence the required root is 8.
Example 2 :
Find the real number root of ∛512
Solution :
Step 1 :
Index of the given square root = 3
Let "x" be the required root
x = ∛512
Step 2 :
In order to remove the square root, raise power 3 on both sides
x³ = (∛512)³
x³ = 512
Step 3 :
Express 512 as the product of two same numbers
That is, 512 = 8 x 8 x 8
512 = 8³
x³ = 8³
Step 4 :
Since the powers are equal, then their base are also equal.
x = 8
Hence the required root is 8.
Example 3 :
Find the real number root of ∛-8000
Solution :
Step 1 :
Index of the given square root = 3
Let "x" be the required root
x = ∛-8000
Step 2 :
In order to remove the square root, raise power 3 on both sides
x³ = (∛-8000)³
x³ = -8000
Step 3 :
Express -8000 as the product of two same numbers
That is, -8000 = (-20) x (-20) x (-20)
-8000 = (-20)³
x³ = (-20)³
Step 4 :
Since the powers are equal, then their base are also equal.
x = -20
Hence the required root is -20.
Example 4 :
Find the real number root of ∜16
Solution :
Step 1 :
Index of the given square root = 4
Let "x" be the required root
x = ∜16
Step 2 :
In order to remove the square root, raise power 3 on both sides
x⁴ = (∜16)⁴
x⁴ = 16
Step 3 :
Express 16 as the product of two same numbers
That is, 16 = 2 x 2 x 2 x 2
16 = 2⁴
x⁴ = 2⁴
Step 4 :
Since the powers are equal, then their base are also equal.
x = 2
Hence the required root is 2.
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