The principle of powers and roots is an integral part of many concepts and topics in mathematics. A thorough understanding of power and roots is essential for learning complex mathematical topics like trigonometry, calculus, polynomials, graphs, etc.

**Power**

A power or an index is a representation of the repeated multiplication of a certain number. Powers are denoted in the form of a^{n} where ‘a’ is the base number and ‘n’ is the power it possesses.

For example, 2^{3 }= 2x2x2 and 3^{4} = 3x3x3x3.

It enables us to break a large number into small fragments of prime numbers and their powers. For example, 7056 = 2^{4}x3^{3}x7^{2} and 1125 = 3^{2}x5^{3}.

A negative power in a number indicates that it divides the other components in that fraction. For example, 2^{-2}=\frac{1}{2*2}=\frac{1}{4} and 3^{2}x 4^{2}x 5^{-2}= \frac{3*3*4*4}{5*5}= \frac{144}{25}.

***NOTE: 0 with negative power is not defined.**

**Properties of Power**

Let’s understand the properties of indices with examples.

**a**^{0}**–**Any non-zero number possessing 0 as power is 1.

a^{0 }= 1. For example, 2^{0}= 1 and 5^{0}= 1

***NOTE: 0**^{0 }**is not defined.**

**a**^{1}**–**Any number comprising 1 as its power is itself.

a^{1}= a. For example, 6^{1}= 6 and 11^{1}= 11

**0**^{n}**–**0 with a certain power can be defined as

0^{n} if n>0 = 0. For example, 0^{9}= 0 and 0^{7}= 0

0^{n} if n<0 =** not defined.**

**1**^{n}**–**1 to the power of any integer is 1.

1^{n}= 1. For example, 1^{23}= 1 and 1^{-23}= 1

**(-a)**^{n}**–**Powers of negative integers can be explained as

(-a)^{n} , if n is even, will give a positive number. For example, (-2)^{4}= 16 and (-5)^{-2}= 125.

(-a)^{n}, if n is odd, will give a negative number. For example, (-7)^{3}= -343 and (-11)^{-1}= -111.

**Mathematical Operations of Powers**

To simplify the equations with numbers containing powers, various mathematical tools are used.

- Entities with the same powers

a^{n}x b^{n}= (ab)^{n}. For example, 2^{2}x3^{2}= (2×3)^{2}= 6^{2}= 36.

a^{n}/b^{n}= ab^{n}. For example, 5^{2}/7^{2}= 57^{2}= 2549.

- Entities with the same base

a^{n}x a^{m}= (a)^{n+m}. For example, 3^{2}x3^{3}= (3)^{2+3}= (3)^{5}= 243

a^{n}/a^{m}= (a)^{n-m}. For example. 5^{3}/5^{2}= (5)^{3-2}= (5)^{1}= 5

(a^{n})^{m}= a^{nm}. For example, (2^{2})^{3}= 2^{2×3}= 2^{6}= 64.

**Roots**

The usage of roots is the inverse operation of powers. Roots are denoted by \sqrt[n]{a} where ‘a’ is the base number and ‘n’ denotes the repeated multiplication a certain number goes through to become ‘a’.

For example, 2^{3}= 8 and \sqrt[3]{8} = 2.

**Square Roots and Cube Roots**

A root in the form of \sqrt[n]{a} where n = 2 is called a square root and a root in the form of \sqrt[n]{a} where n = 3 is called a cube root. Generally, square roots are showcased without mentioning the ‘n’ component of the root form, i.e, \sqrt{8}.

For example, \sqrt{49} = 7 and \sqrt{121} = 11.

These are the most common roots that are used across different disciplines of mathematics.

**Properties of Roots**

Following are the properties that regulate the functions of roots.

- \sqrt[n]{a} x \sqrt[n]{b} = \sqrt[n]{ab} . For example,
**\sqrt[3]{1331}**x**\sqrt[3]{343}**=**\sqrt[3]{1331*343}**= 11×7 = 77. - nanb =
**\sqrt[n]{\frac{a}{b}}**. For example, 481416 == \frac{3}{2} = 1.5**\sqrt[4]{\frac{81}{16}}** - a^{\frac{1}{n}} = \sqrt[n]{a} . For example, 4
^{12}= \sqrt{4} = 2. - \sqrt[n]{a}+\sqrt[n]{b} \sqrt[n]{a+b} . For example, \sqrt[3]{27}+\sqrt[3]{125} because \sqrt[3]{27} = 3 and \sqrt[3]{125} = 5 and 3+5 = 8, but \sqrt[3]{27+125} i.e \sqrt[3]{152} \neq 8.

***NOTE: negative roots (complex numbers) are not included in the syllabus of GMAT.**

**Sample Question**

If 0<m<1, then which of the following must be true?

- _{m}2>\frac{1}{m}> \sqrt{m}
- \frac{1}{m}> _{m}2 > \sqrt{m}
- \frac{1}{m}> \sqrt{m} > _{m}2
- \sqrt{m} > _{m}2 > \frac{1}{m}
- \sqrt{m} > \frac{1}{m}> _{m}2

**Answer: C**

**Explanation:**

Plug in the value for *m *and check the answer options.

m=\frac{1}{2}

then

_{m}2=\frac{1}{4}=0.25 , \sqrt{m}=\frac{1}{\sqrt{6}}≈0.7 and \frac{1}{m}=\frac{1}{\frac{1}{2}}=2

So, then the correct ordering from greatest to least is \frac{1}{m} > \sqrt{6} > _{m}2

**Hence, the answer is C.**