Root Calculator

Root is the number obtained when a number is raised to a fractional power. In mathematics, the nth root of a number ‘x’ is another number ‘y’. This means that, when ‘y’ is multiplied by itself n times, the result is ‘x’. The mathematical notation for this is $$ {n\sqrt x} = y $$. The number ‘n’ is also called the index and the number ‘x’ is called the radicand.

The symbol used to denote roots is √ which is called the radical sign. The two roots which are mostly commonly used are square root and cube root. For square root, the notation is √x = y and for cube root, the notation is ³√x=z. For any other root apart from the square root, the index ‘n’ is explicitly mentioned.

The formula for any root is $$ {n\sqrt x} = x^{1\over n} $$

The n^th root of a number ‘x’ indicates that ‘x’ has been raised to the power 1/n. In the above equation, the left-hand side is called the radical notation and the right side is called the exponent form.

Therefore, the square root is √x = x^1/2, the cube root is ³√x = x^1/3 and any other root is ⁿ√x = x^1/n. $$ {n\sqrt x} = y $$

This formula indicates that the number ‘y’ needs to be multiplied with itself ‘n’ times to obtain ‘x’.

Consider the number 64. The square root is √64 = 64^1/2 = ±8, because both 8² = 64 and (−8)² = 64. The square root is ³√64 􀰯 = 64^1/3 = 4, because 4³ = 64. In this case, (−4)³ ≠ 64. The 6^th root of 64 is ⁶√64 = 64^1/6 = ±2, because both 2⁶ = 64 and (−2)⁶ = 64.

It is also possible to find the roots of numbers with decimal points. For example, the square root of 0.64 is √0.64 = ±0.8, and the cube root is ³√0.64 = 0.86. The 6^th root of 0.64 is ⁶√0.64 = ±0.928. It is not possible to find roots if the radicand is negative. Such roots result in imaginary numbers.

When the index ‘n’ is even, the resulting number ‘y’ can be both positive and negative. When n is odd, the resulting number ‘y’ is always positive.

Given below are some rules for finding roots.

1) In the usual notation, the number ‘y’ needs to be multiplied with itself ‘n’ times to obtain ‘x’ $$ {\sqrt[n] x} = y \; or \; y^n = x $$

    For example, $$ {\sqrt x} = y \; or \; y^2 = x $$

2) The n^th root of a number xn is x itself. In other words, $$ {\sqrt[n] {x^n}} = x^{n\over n} = x $$

    For example, $$ {\sqrt[3] {27^3}} = 27^{3\over 3} = 27 $$

3) The above rule can be generalized for cases when n is even or odd. $$ {\sqrt[n] {x^n}} = |x| \;when \;n \;is \;even. \\ {\sqrt[n] {x^n}} = x \;when \;n \;is \;odd. $$

    For example, $$ {\sqrt[7]{9^7}} = 9 \; and \; {\sqrt[8] {9^8}} = |9|. $$     This is because, both (−9)⁸ and (9)⁸ give the same result.

4) When there are two or more roots which are getting multiplied, with the same index but different radicands, then we can use only one root, and multiply the different radicands within the same root. $$ {\sqrt[n] x} * {\sqrt[n] y} = {\sqrt[n] {xy}} $$

    For example, $$ {\sqrt[3] 4} * {\sqrt[3] 16} = {\sqrt[3] {4*16}} = {\sqrt[3] 64} $$

5) When there are two or more roots which are getting divided, with the same index but different radicands, then we can use only one root, and divide the different radicands within the same root. $$ {{\sqrt[n] x} \over {\sqrt[n] y}} = {\sqrt[n] {x\over y}} $$

    For example, $$ {{\sqrt[3] 16} \over {\sqrt[3] 4}} = {\sqrt[3] {16\over4}} $$

6) If the root is being raised to the power ‘m’, then the reciprocal of the index ‘n’ of the root can be multiplied with the power ‘m’. $$ ({\sqrt[n]x})^m = (x^{1\over n})^m = x^{m\over n} $$

    For example, $$ ({\sqrt[3]8})^2 = (8^{1\over 3})^2 = 8^{2\over 3} $$

This rule works like having an exponent raised to another exponent. In that case, the two exponents get multiplied.

7) Basic arithmetic operations like, addition and subtraction can be performed on two or more roots only if the radicand is the same. Here, only the coefficients of the roots get added or subtracted. $$ {a{\sqrt[n]x}} + {b{\sqrt[n]x}} = (a+b){\sqrt[n]x} \; and \; {a{\sqrt[n]x}} - {b{\sqrt[n]x}} \; = \; (a-b){\sqrt[n]x} $$

    For example, $$ {3{\sqrt[5]64}}+{7{\sqrt[5]64}} = (3+7){\sqrt[5]64} = {10{\sqrt[5]64}} \\ 5{\sqrt2}-4{\sqrt2}=(5-4){\sqrt2} = {\sqrt2}$$

8) If the number ‘x’ is an even perfect square, then the square root of that number is always even. Similarly, if the number ‘x’ is an odd perfect square, then the square root of that number is odd.

    For example, $$ {\sqrt16} = 4 \; and \; {\sqrt25}=5. $$

9) If the radicand is a negative number, then the root is said to be imaginary. $$ {\sqrt[n]{-x}} \,= \, imaginary \;number $$

    For example, $$ {\sqrt{-16}} \;and \;{\sqrt[3]{-9}} \; are \; imaginary \; numbers $$

The root of a number can be computed by splitting it up into smaller numbers. The steps for this are described below.

    i) Split up the radicand inside the root by writing out all the prime factors.

    ii) If the index is ‘n’, and there is a number which occurs within the root ‘n’ times, pull out only one instance of that number and write it in front of the radical sign.

    iii) If the root has a coefficient, then the number which is pulled out gets multiplied with the coefficient.

    iv) If any number does not get repeated ‘n’ times within the radical sign, keep the number as it is.

Consider the following example of simplifying a root. $$ {\sqrt[3]7}*{\sqrt[3]8} $$

This can be written as ³√(7x8) using the product rule. Now we list all the prime factors of the numbers. $$ {\sqrt[3]{7*2*2*2}} $$

Since 7 is a prime number, it remains unchanged. Also, the number 2 occurs three times which is equal to the index 3, so only one 2 is pulled out. The resulting simplified root is shown below. $$ 2{\sqrt[3]7} $$

Some of the applications of roots in real life are as follows:

1) They are used extensively in mathematics and electrical engineering to find fractional exponents of certain numbers.

2) They are used in the calculation of compound interest.

3) They are used in biological measurements to compare the sizes of different animals.

4) In finance, roots are used in the calculation of interest and depreciation.

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