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Exponents

Last Updated : 24 Jan, 2024
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Exponents are the basic concept used in mathematics that are helpful in solving and understanding very large numbers. Suppose we have to simplify a very large number such as 10 multiplied by itself 10 times then the number is represented as, 1010 which is a very easy way of representing the numbers. Exponent is also called the power of a number. The exponent of the number can be integers or fractions, The fraction exponent is also called the radical.

What are Exponents?

When any number or variable (x) is multiplied n times, then the resultant is xn. Then, n is called the exponent of 

x.x.x.x.x.x.x … n times = xn

Then x is base and n is an exponent of that base and this is read as x raises to power n. An exponent is the power of a number that indicates the number of times the number multiplied by itself. Exponent defines the number of times a number is multiplied by itself. 

Example: 2.2.2 = 23

  • Base = 2
  • Exponent = 3

The concept of the exponent is represented in the image added below,

Exponet of a Number

Exponents Definition

A mathematical way of representing very large numbers is called the exponents. It is a way of representing the powers of a number. Suppose we have a number 5 which is multiplied by itself 5 times then this is expressed as, 5×5×5×5×5

As we can see this is a very tedious way of representing the number in exponent this is represented as,

5×5×5×5×5 = 55

Thus, the exponent is a very short way of representing large numbers.

Exponents Formulas

The formulas that are widely used for solving the exponents are added in the table below,

  Exponents Formulas

n times product exponent formula x.x.x.x … n times = xn   
Multiplication Rule xm . xn = x(m + n) 
Division Rule xm/xn = x(m – n)
Power of product rule (xy)n = xn. yn
Power of fraction rule (x/y)n = xn/yn 
Power of power rule [(x)m]n = xmn
Zero Exponent (x)0 = 1, if x ≠ 0
One Exponent (x)1 = x
Negative Exponent x-n = 1/xn
Fractional Exponent xm/n = n√(x)m

Note: If an equation base is the same we can equate the exponents.

Laws of Exponents

There are seven laws of exponents that we study under this heading.

Product of Power Rule: This rule states that two numbers in exponential having the same base are multiplied then their product contains the same base and their powers get added. For Example 23⨯24 = 23+4 = 27

Quotient of Power Rule: This rule states that two numbers in exponential form is divided then the quotient has the same base and their powers get subtracted. For Example: 35/32 = 35-2 = 33

Power of Power Rule: If a number in exponential form is raised to some power then its powers get multiplied. For Example, (43)2 = 43⨯2 = 46

Power of a Product Rule: If two numbers in the exponential form that have the different bases but the same exponents are multiplied then the product has the base equal to the product of two bases and the power remains the same. For Example, 32⨯42 = 122 = 144

Power of Quotient Rule: If two numbers in the exponential form that have a different base but the same exponents are divided then the quotient has the base equal to the quotient of two bases and the power remains the same. For Example, 63/33 = 23 = 8

Zero Exponent Rule: Any number raised to power zero gives 1. For Example, (101)0 = 1

Negative Exponent Rule: If any number is raised to negative power then to make the power positive, the base is converted to its reciprocal. For Example, 2-3 = (1/2)3 = 1/23 = 1/8

Negative Exponents

Negative Exponent is nothing but the exponents of the reciprocal numbers thus, negative exponents are easily solved by taking the reciprocal and then easily solving the exponent using the normal rules. This is represented as,

x-n = (1/x)n

Suppose we have to solve for the negative exponent (2)-3 then,

(2)-3 = (1/2)3 = 1/8

Thus, taking the exponent easily solves the exponents. All the formulas of the exponents work easily with the negative exponents.

Exponents with Fractions

The exponents with the fraction are also called the radicals. These are the exponents that have a fraction of their power. The square root, cube root, nth root, and others all are called exponents with fractions.

We represent the fraction exponents as,

  • Square Root = √()
  • Cube Root = 3√()
  • nth Root = n√()

Now the fraction exponent is solved in two parts in the first part we solve the denominator and then solve the numerator, this is represented as,

xn/m  = {(x)1/m}n

here, we first solve (x)1/m and then take its nth power to get the final answer. This can be understood by the example added below,

Example: Simplify 43/2

Solution:

= 43/2

= (41/2)3

= 23 = 8

Decimal Exponents

Decimal Exponents are nothing but the other way of representing the fraction exponents. If any exponent is given in the decimal form then we first change it into fraction form and then easily solve for the fraction form.

This can be understood by the example added below,

Example: Simplify 41.5

Solution:

= 41.5    (As, 1.5 = 3/2)

= 43/2

= (41/2)3

= 23 = 8

Exponent Table

Type of Exponent Expression Expansion Simplified value
Zero exponent 60 1 1
One exponent 41 4 4
Exponent and power 23 2 × 2 × 2 8
Negative exponent 5-3 1/53 = 1/(5 × 5 × 5) 1/125
Rational exponent 91/2 √9 3
Multiplication 32 × 33 3(2 + 3) = 35 273
Quotient 75/ 73 7(5 – 3) = 72 49
Power of exponent (82)2 8(2 × 2) = 84 4096

Scientific Notation with Exponents

Scientific Notation is a way of writing very large numbers into very small numbers. In scientific notation, the numbers are represented in the multiple of 10. The number is first converted into its unit form and then the number is multiplied with the power of 10 to get the number in scientific notation.

These numbers are useful in writing very large and very small numbers. Suppose we have to write 15670000 then in scientific notation it is represented as, 1.567×107

Any number can be easily represented in the scientific notation by following the steps added below,

  • Step 1: If the number is greater than one mark the decimal digit after the first digit from the starting of the number. 
  • Step 2: Then multiply the number with the 10 raise to the power as their are digits after the decimal or point (include zero in the counting)
  • Step 3: If the number is smaller than one shift the decimal to the first digit counting from the left of the number excluding zeros.
  • Step 4: Then multiply the number with the 10 raise to the negative power as their are digits from which the decimal is shift.

For example, Convert 134500000000 into scientific notation.

Solution:

= 134500000000

= 1.345 × 108

For example, Convert 0.0000001345 into scientific notation.

Solution:

= 0.0000001345

= 1.345 × 10-7

Also, Read

Exponents Examples

Example 1: Solve the following: 

  1. 2.2.2.2     
  2. 32.33  
  3. (4.5) 
  4. (5) 
  5. 2-2 
  6. 25/23 
  7. [(3)1]
  8. 43/2 
  9. (4/3)

Solution: 

  1. 2.2.2.2 = 24 =16
  2. 32.33 = 3(2 + 3) = 35 = 243
  3. (4.5)2 = 42.52 = (16).(25) = 400
  4. (5)0 = 1
  5. 2-2 = 1/22 = 1/4
  6. 25/23 = 2(5-3) = 22 = 4
  7. [(3)1]2 = 3(1.2) = 32 = 9
  8. 43/2 = √(4)3 = √64 = 8
  9. (4/3)2 = 42/32 = 16/9

Example 2: Simplify: 

  1. (23 ÷ 24)-2.23      
  2. 3(-2)÷ 42   
  3. 33.42/64   
  4. (3-1 + 2-2 + 4-1)

Solution: 

(1)

(23 ÷ 24)-2.23 

= (23/24)-2.23 

= [2(3 – 4)]-2.23 

= [2-1]-2.23 

= 2(-1).(-2).23 

= 22.2

= 25 = 32

(2)

3(-2) ÷ 4

= 1/(3)2(4)

= 1/9.16 = 1/144

(3)

33.42/64 

= 33.42/(2.3)4 

= 33.24/24.34 

= 1/3

(4)

(3-1 + 2-2 + 4-1

= (1/3 + 1/22 +1/4) 

= (1/3 + 1/4 + 1/4) 

= 5/6

Example 3: Find the value of x if (4)x + 12 = (4)2x + 6.(2)6

Solution:  

(4)x+12 = (4)2x+6.(22)3

(4)x+12 = (4)2x+6.(4)3

(4)x+12 = (4)2x+6+3

(4)x+12 = (4)2x+9

Since, bases are equal powers gets equated

x +12 = 2x + 9

2x – x = 12 – 9

x = 3

Example 4: Find the value of {3434/3}1/4

Solution: 

{3434/3}1/4  = {(73)4/3}1/4  

= {7}3.(4/3).(1/4) = 7

Example 5: Find the value of x + y if:

(81)y = 27/(3)x, 4y= 256

Solution: 

(34)y = (33)/(3)x

(3)4y = (3)3-x

Since, bases are equal then powers get equated

4y = 3-x ⇢ Equation (1)

4y = 256

4y = (4)4

y = 4

Putting the value of y in Equation 1, 

4.4 = 3-x

16 = 3-x

x = -13

Now, we have to find value of x + y

x + y = -13+4 = -9

Example 6: If (-9)2x+7 = (-9)x. 81, then find the value of (x2 + 1)/(x2 – 12).

Solution: 

(-9)2x+7 = (-9)x . 81

(-9)2x+7 = (-9)x . (-9)2

(-9)2x+7 = (-9)x+2

Since, bases are equal then powers get equated

2x + 7 = x + 2

2x – x = 2 – 7

x = -5

Now, we have to find value of  (x2 + 1)/(x2 – 12) 

(x2 + 1)/(x2 – 12)  = [(-5)2 + 1]/[(-5)2 – 12]

= [25 + 1]/[25 – 12]

= 26/13  

(x2 + 1)/(x2 – 12) = 2

Example 7: Find multiplicative inverse of [(-13)-1]2 ÷ (91)-1

Solution: 

Let, x = [(-13)-1]2 ÷ (91)-1 

x = (-13)-2 ÷ (91)-1 

= (-1/132) ÷ (1/91)

= (-1/132) × 91

x = -7/13

Multiplicative inverse is given by 1/x i.e.

1/x = 1/(-7/13)

1/x = -13/7

Practice Questions on Exponents

Q1: Solve (32)1/5 +(-9)0 + (64)1/3

Q2: Simplify (64/81)-3/4 ⨯ (25/9)-3/2

Q3: Find x when 3x+2/35 = 27

Q4: Find x when (64)x = 16/4x

FAQs on Exponents

1. What are Exponents in Math?

The exponents in mathematics are a way of representing very large and very small numbers they are used in mathematics for simplifying and easing the calculation.

2. What are the Examples of Exponents?

Some example of the exponents are,

  • 3.456×10-12
  • 1.4526×1022
  • 9.87×10-21, etc.

3. How to Multiply Exponents?

To multiply exponents we use the following formula,

  • an × am = am+n
  • an × bn = (ab)n

This can be further understood using the example,

Example: Simplify 33 × 38

Solution:

= 33 × 38

= 33+8 = 311

4. What is a Zero Exponent?

A zero exponent is an exponent with power 0 and the value of the zero exponent is always zero, i.e.

a0 = 1 (a ≠ 0)

5. What is a Negative Exponent?

A negative exponent is an exponent with negative values and its value is calculated by taking its reciprocal and them making its exponent positive, then simplifying it normally, i.e.

a-n = (1/a)n

6. What is 2 with an Exponent of 4?

The value of 2 with an exponent 4 is,

= 24

= 2×2×2×2

= 16



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