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} : Any non-zero number possessing 0 as power is 1.

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

*NOTE: 00 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)^{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?

  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=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.

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