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} x 3^{3} x 7^{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\times 2} = \frac{1}{4} and 3^{2}x 4^{2}x 5^{-2}= \frac{3\times 3\times 4\times 4}{5\times 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} :**

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**^{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)^{0} , if n is even, will give a positive number. For example, (-2)^{4} = 16 and (-5)^{-2} = \frac{1}{25} .

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

**Mathematical Operations of Powers**

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

- Entities with the same powers

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

a^{n} \div b^{n} = (\frac{a}{b})^{n}. For example, 5^{2} /7^{2} = (\frac{5}{7})^{2}= \frac{25}{49} .

- Entities with the same base

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

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

(a^{n} )^{m} = a^{nm} . For example, (2^{2} )^{3} = 2^{2\times3} = 2^{6} = 64.

**Roots**

The usage of roots is the inverse operation of powers. Roots are denoted by na 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]{2} = 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[]{a} .

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} \times \sqrt[3]{343} = \sqrt[3]{1331 \times 343} = 11×7 = 77.

- \frac {\sqrt [n] {a} } {\sqrt [n] {b} } = \sqrt [n] {\frac {a} {b} } . For example, \frac { \sqrt [4] {81} }{ \sqrt[4] {16}} = \sqrt[4]{\frac{81}{16}} = \frac{3}{2} = 1.5

- a^{\frac{1}{n}} = a\sqrt[n]{a}. For example, 4^{\frac{1}{2}} = \sqrt{4} = 2.

- \sqrt[n]{a} +\sqrt[n]{b} \neq \sqrt[n]{a+b} . For example, \sqrt[3]{27} +\sqrt[3]{125} \neq \sqrt[3]{27+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=12

then

m ^{2} = \frac{1}{4} = 0.25 , \sqrt{m} =\frac{1}{\sqrt{2}} ≈ 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{m} > m ^{2}

**Hence, the answer is C.**