[Maths Class Notes] on Exponents and Powers Formulas Pdf for Exam

If you wonder time and again 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 number 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 = a(mn)

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

What is Power in Mathematics?

In Mathematics, Power is defined as an expression that represents repeated multiplication of the same given number. It is written as ‘raising a number to the power of any other number’.

For instance, 7 × 7 × 7 × 7 = 2041, which can also be written as  74 = 2041, which means the number ‘7’ is to be multiplied four times by itself to get the number ‘2041’. In other words, it can be said that the number ‘4’ raised to the power of 4 or the number ‘7’ raised to the 4th power” thereby giving us the number ‘7’. Here, the number ‘4’ is called the base number and ‘4’ is known as the power or exponent.

What is an Exponent in Mathematics?

Exponent in Mathematics is defined as a positive or negative number which describes the power to which the base number is raised. In other words, it states the number of times a number needs to be used in a multiplication process.

For instance, 63 = 6 × 6 × 6 equals 216. Where, the base number is equal to ‘6’ which is used for 3 times in a multiplication. Hence, we are multiplying the number ‘6’ three times by itself to get the number ‘216’. In geometry, cube and square are the two most commonly used exponents.

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″.

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^{(ytimes z)}]

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

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

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

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

[sqrt{a}+sqrt{b}] ≠ [sqrt{(a+b)}]

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

 Exponent and Radical Rules Example Notes xᵐ=x⋅x⋅x⋅x…..(m times) 2³=2⋅2⋅2=8 x is the base, m is the exponent [sqrt[m]{x}=y]  means  [y^m=x] [sqrt[3]{8}=2],  since 2⋅2⋅2=2³=8 y is the base, m is the index (root) [sqrt{x}] means  [sqrt[2]{x}] [sqrt{4}=2]  means  [sqrt[2]{4}=2] The default root is 2 (square root). [x^{frac{m}{n}}=(sqrt[n]{x})^m=sqrt[n]{x^m}] [x^{frac{2}{3}}=sqrt[3]{8^2}=(sqrt[3]{8})^2] The numerator of the exponent is the power and the denominator is the root if a root is raised to a rational fraction. The exponent can be on the inside or outside when raising a radical to an exponent. [x^{-m}=frac{1}{x^m}][frac{1}{x^{-m}}=x^m] [left ( frac{x}{y} right )^{-m}=left ( frac{y}{x} right )^m] [2^{-2}=frac{1}{2^2}=frac{1}{4}] [frac{1}{2^{-2}}=2^2=4] [left ( frac{2}{3} right )^{-2}=left ( frac{3}{2} right )^2=frac{9}{4}] If a base is raised to a negative exponent, it means that taking the reciprocal and making the exponent positive. [asqrt{x}times bsqrt{y}=absqrt{xy}] [2sqrt{3}times 4sqrt{5}=8sqrt{15}] By multiplying two radical terms, you can multiply what is on the outside (coefficient), and also what’s on the inside (radicand). It can only be done if the roots (indices) are the same as square root or cube root. [sqrt[textrm{even}]{textrm{negative numbers}}] exists for imaginary numbers, but not for real numbers. [sqrt[4]{-16}] = no real solution The even root of a negative number can’t be taken to get a real number. However, an imaginary number can be achieved. [sqrt[textrm{odd}]{x^{textrm{odd}}}=x] [sqrt[textrm{even}]{x^{textrm{even}}}=mid xmid ] [sqrt[3]{(-2)^3}=sqrt[3]{-8}=-2] [sqrt{(-2)^2}=sqrt{4}=2] A root undoes raising a number to that exponent. For instance, under the square root we squared  –2, but our answer we got is 2, which is |−2| (the absolute value of 2. For [y=x^{textrm{even}},y=pm sqrt[textrm{even}]{x}] [x^2=16; x=pm 4] It is required to include both the positive and negative solutions in an equation with an even exponent when taking an even root. Since both the positive root and negative roots work when raised to  that even power. The square root sign only gives the positive solutions.

Solving Exponential Equations using Laws

As now you are already aware that you 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.

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

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

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

[4x^2 2x^3 8x^{-4}=2^{(3)^2}]

[4times 2times 8times {x^2x^3x^{-4}}=2^6]

[4times 2times 8times x^{2+3-4}]=64

[8times 8times x^1=64]

[64x=64]

Thus, [x=1]

2.Multiplication of Exponent Rule

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

[(frac{x}{y})^n=frac{x^n}{y^n}]

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

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

[frac{x^4}{y^3}divfrac{y^5}{x^2}=frac{1}{4}]

3.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.

[4x^4] + 8 = 72

Isolate the term [x^4] by subtracting 8 from both sides and then divide both sides by 4.

[4x^4] = 72 – 8

[x^4] = 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).

[x^4] = 16

{[x^4]}(¼)= 16(¼)

Hence, we get [x^4] = 2

4.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.

[2x^2=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.

[x^2-x+6=8]         ….……2

[x^2-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

→ (ym-n)m-n×(yn-o)n-o×(yo-m)o-m

Using the formula

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

→y(m²+n²−2mn)×y(m²+o²−2no)×y(o²+m²−2om)

Doing the Math

ma.mb=ma+b

→ y(m²+n²−2mn+n²+o²−2no+o²+m²−2om)

→ y2(m²+n²+o²−(mn+no+om))

→ y[2(0)]

→ y0=1

Example 2: Find y if

32y-1+32y+1=270

Solution2:

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

→ 32y-1(1+32)

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.

→ 32y-1(10)=270

→ 32y-1=27

→ 32y-1=33

→ 2y−1=3

→ y = 2.