Power and Roots

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 aⁿ where ‘a’ is the base number and ‘n’ is the power it possesses. 

For example, 2³ = 2x2x2 and 3⁴ = 3x3x3x3. 

It enables us to break a large number into small fragments of prime numbers and their powers. For example, 7056 = 2⁴x3³x7² and 1125 = 3²x5³. 

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 3²x 4²x 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.

1. a⁰– Any non-zero number possessing 0 as power is 1.

a⁰ = 1. For example, 2⁰= 1 and 5⁰= 1 

*NOTE: 0⁰ is not defined.

2. a¹– Any number comprising 1 as its power is itself.

a¹= a. For example, 6¹= 6 and 11¹= 11

3. 0ⁿ– 0 with a certain power can be defined as

0ⁿ if n>0 = 0. For example, 0⁹= 0 and 0⁷= 0 

0ⁿ if n<0 = not defined.

4. 1ⁿ– 1 to the power of any integer is 1.

1ⁿ= 1. For example, 1²³= 1 and 1^-23= 1

5. (-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

aⁿx bⁿ= (ab)ⁿ. For example, 2²x3²= (2×3)²= 6²= 36.

aⁿ/bⁿ= abⁿ. For example, 5²/7²= 57²= 2549.

• Entities with the same base

aⁿx a^m= (a)^n+m. For example, 3²x3³= (3)²⁺³= (3)⁵= 243

aⁿ/a^m= (a)^n-m. For example. 5³/5²= (5)^3-2= (5)¹= 5

(aⁿ)^m= a^nm. For example, (2²)³= 2^2×3= 2⁶= 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, 2³= 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.

1. \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.
2.  nanb = \sqrt[n]{\frac{a}{b}} . For example, 481416 = \sqrt[4]{\frac{81}{16}} = \frac{3}{2} = 1.5
3. a^{\frac{1}{n}} = \sqrt[n]{a} . For example, 4¹² = \sqrt{4} = 2.
4. \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?

1. _{m}2>\frac{1}{m}> \sqrt{m}
2. \frac{1}{m}​> _{m}2 > \sqrt{m}
3. \frac{1}{m}​> \sqrt{m} > _{m}2
4. \sqrt{m} > _{m}2 > \frac{1}{m}​
5. \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.