Exponents and Powers Formulas

If you time and again wonder if what is an exponent in math, then know an Exponent, also known as power, is a mathematical mannerism of expressing a number multiplied by itself by a certain definite of times. In mathematical terms, when we write a non integer number a, it is actually a1, referred as a — to the power 1.

a2 = a*a

a3 = a*a*a

a4 = a*a*a*a

a5 = a*a*a*a

:

:

an = a*a*a*a*a*a*a . . . n times.


The following is the standard formula of exponent and power to solve problems on exponents.

(am)n = (an)m = amn


The given chart splits the parts in an exponential expression, defining exactly which number is the exponential power to the factor.


Tabular Representation Clarifying Definitions

Expressions

Long Hand Expression

Base

Exponent/Power

Value

25

2×2×2×2×2

2

5

32

53

5×5×5

5

3

125

35

3×3×3×3×3

3

5

243

74

7×7×7×7

7

4

2401


In reading problems, mathematical expressions with exponential powers such as74 are often pronounced "seven to the fourth power." Then again, exponential expressions such as 74 are often read as "the 4th power of 7".


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Laws of Exponents Formulas

There are various laws of exponents that you should practice and remember in order to thoroughly understand the exponential concepts. The following exponent law is detailed with examples on exponential powers and radicals and roots.

\[\frac{x^{n}}{x^{m}}= x^{n - m}\]

\[x^{n}. x^{m} = x^{n + m}\]

\[x^{n} y^{n} = (xy)^{n}\]

\[\frac{x^{n}}{y^{n}} = \frac{x^{n}}{y^{n}}\]

\[x^{-n} = \frac{1}{x^{-n}}\]

\[(x^{y})^{z} = x^{(y \times z)}\]

\[\sqrt{x} \times \sqrt{x} = \sqrt{x^{n}}\]

\[\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}\]

\[[\sqrt{x^{n}}] = \sqrt{x^{n}}\]

\[a\sqrt{c} + b\sqrt{c} = (a + b) \sqrt{c}\]

\[\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}\]

\[\frac{ax}{y} = \sqrt[y]{ax}\]


Solving Exponential Equations Using Laws

As now you are already aware that possess one or more terms with a base that is raised to a power ≠ 1. While there is no definite formula for solving an exponential equation, the following rule will help you clearly understand different ways of finding the unknown value in an exponential equation.

  1. Multiplication of Exponent Rule

To solve exponential equations, the following are the most important formulas that can be used to multiply the exponents together.

xn xm = x n+m

xn yn = (xy)n

Now taking the following example with the above power and exponent formula:

4x2 2x3 8x-4 = (23)2

4×2×8× {x2x3x-4} = 26

4×2×8× x2+3-4 = 64

8 ×8 ×x1 = 64

64  x = 64

Thus, x = 1

  1. Multiplication of Exponent Rule

The given formulas are applicable in dividing exponents, particularly these two formulas.

{x/ y}n =xn / yn

x-n = 1/ xn

Now taking the following example with the above exponent formula will help understanding how to divide exponents.

X4/ y3/ y5/ x2 = ¼

  1. Isolation and Raise to the Inverse Exponent Rule

We have to organize the term with an exponent on one side while the other terms on the other side of the equation. Next is to raise both sides of the equation to the inverse exponent.

4x4 + 8 = 72

Isolate the term x4 by subtracting 8 from both sides and then divide both sides by 4.

4x4 = 72 - 8

x4 = 16

Now, to isolate x, since x is raised to 4/1th power, raise both sides of the exponential equation to the inverse power (1/4).

x4 = 16

{x4}(¼)= 16(¼)

Hence, we get x4 = 2

  1. Factorization

Solving an equation by isolating an exponent makes it easier to deal with exponent problems. You have to arrange all identical terms on one side of the (=) sign and then factor it.

2x2 = 2x + 12 = 16    ……. 1

At this step, divide each term by common factor (i.e. 2) and then subtract the number on the right side.

x2 - x + 6 = 8         ………2

x2 - x – 2 = 0         ……….3

Apply rule of factoring, simplify and solve

We obtain

(x-2) (x+1)

X= +2, -1.


Solved Examples of Exponents and Powers Formulas

Example1:

If m2+n2+o2 = mn+no+om, simplify [ym/yn]m-n × [yn/yo]n-o × [yo/ym]o-m


Solution1:

Using ma/nb = ma-b, we obtain

→ \[(y^{m-n})^{m-n} \times (y^{n-o})^{n-o} \times (y^{o-m})^{o-m}\]

Using the formula

(m-n)2 = m2+n2-2mn in the exponent,

→ \[y^{(m^{2} + n^{2} - 2mn)} \times y^{(m^{2} + o^{2} - 2no)} \times y^{(o^{2} + m^{2} - 2om)}\]

Doing the math

\[m^{a}. m^{b} = m^{a + b}\]

→ \[y^{(m^{2} + n^{2} - 2mn + n^{2} + o^{2} - 2no + o^{2} + m^{2} - 2om)}\]

→ \[y^{2(m^{2} + n^{2} + o^{2} - (mn + no + om))}\]

→ \[y^{[2(0)]}\]

→ \[y^{0}= 1\].

 

Example 2: Find y if \[3^{2y - 1} + 3^{2y + 1} = 270\]


Solution2:

We will first take out a term in common, with which we obtain

→ \[3^{2y - 1}(1 + 3^{2})\]

See that here, we are using the formula for any non-integer am+n = am. an in expressing 32y+1 as a product of 32y-1 and 32.

→ \[3^{2y - 1} (10) = 270\]

→ \[3^{2y - 1} = 27\]

→ \[3^{2y - 1} = 3^{3}\]

→ \[2y - 1 = 3\]

→ y = 2.

FAQ (Frequently Asked Questions)

1. What is Power and Exponent?

Ans.

  1. Base - the number which is multiplied by itself a certain number of times and is used as a factor is called a base. For example: in the expression 54, the number 5 is called the base, and the number 4 is called the exponent or power. The exponent tally perfectly to the number of times the base is used as a factor.

  2. Exponent - We exactly know how to calculate the expression 3 x 3. This expression can also be written in a shorter way using something called exponents. 3.3 = 32. Here in the expression 2 is the exponent. Thus, the number of times a value is multiplied by itself is what we call an exponent.

  3. Power - It is actually a synonym for exponent, which describes an exponential expression that’s simply multiplication repeated. For example, if we are to say, "Raise 4 to the 7th power," the base here would be 4 and the power (or exponent to which 4 is raised) would be 7.

2. What is Meant by Radical and Root in an Exponential Equation?

Ans. Radical is a mathematical symbol that is used to denote the square or nth root of a number. As an example, the value of "radical 16" is 4 and the value of "radical 64" is 8.