Exponents Formula

Exponent is one of the important basics of mathematics. Exponents are used in various formulas like series, binomial expansion, and many more. Various exponent formulas are used in the various fields of mathematics. Exponent formulas are very easy and useful. Let’s learn about exponents and their formulas. Exponents are the powers on any variable or constants. 

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. An exponent is a power on a number. It is multiplied by itself. Exponent defines the number of times a number is multiplied by itself. Example, 2.2.2 = 2³, base = 2, exponent = 3.

Exponents Formulae

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.

Sample Questions 

Question 1: Solve the following: 

• 2.2.2.2     
• 3².3³
• (4.5)² 
• (5)⁰ 
• 2^-2
• 2⁵/2³
• [(3)¹]² 
• 4^3/2
• (4/3)² 

Solution: 

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

Question 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 = 2².2³ = 2⁵ = 32
2. 3^(-2) ÷ 4² = 1/(3)²(4)² = 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

Question 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

Question 4: Find the value of {343^4/3}^1/4

Solution: 

{343^4/3}^1/4 = {(7³)^4/3}^1/4 

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

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

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

4^y = 256

4^y = (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

Question 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

Question 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

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