Exponent Rules

What are exponents?

For any real number “ a”  and a positive integer “ n”, we define a ⁿ  as

a ⁿ =  a x a x a x a x a . . . . . . . . . .( n times ).

Here a ⁿ is called the nth power of a. the real number a is called the base and n is called the exponent of the nth power of a.

The explanations and examples below on exponent rules follow on from the Power (Exponents and Bases) page which you might want to start with.

• Product Rule
• Zero as an Exponent
• Quotient Rule
• Negative Exponents

The examples on simplifying exponential expressions show when and how the above exponent rules are used.

Rules of Exponents

Rules of exponents are similar to laws of integer exponents. Let us understand some of them in further detail.

Negative Integral Power

We use the negative exponent rule to change an expression with a negative exponent to an equivalent expression with a positive exponent. The rule states that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power. In other words, an expression raised to a negative exponent is equal to 1 divided by the expression with the sign of the exponent changed.

Hence, for any non – zero real number ‘ a ‘ and a positive integer n, we define.

a ^{– n} = $\frac{1}{a^n}$

Let us understand it through some examples.

5 ^{– 3} = $\frac{1}{5^3} = \frac{1}{5 x 5 x 5} = \frac{1}{125}$

$( \frac{3}{2} )^{-2} = \frac{1}{( \frac{3}{2} )^2} = \frac{1}{\frac{3}{2}x\frac{3}{2}x\frac{3}{2}} = \frac{1}{\frac{27}{8}} = \frac{8}{27}$

The Product Rule

The product rule for exponents states that when we multiply exponential expressions having the same base, we can add the exponents and keep the base unchanged. This means that

a ^m x a  ⁿ =  ( a x a x a x a . . . . . . . . . . . . to m factors ) x ( a x a x a x a x . . . . . . . . . to n factors )

  =  a x a x a x a x a x . . . . . . . . . . . . . . . to ( m + n ) factors

  = a ^{m + n} 

Hence, a ^m x a  ⁿ= a ^{m + n}

Let us understand it using some examples.

5 ⁴ x 5 ³ = 5 ^{4 + 3} = 5 ⁷ 

4 ⁸ x 4 ⁶ = 4 ^{8 + 6} = 4 ¹⁴

The above rule can be summarised as –

Product Rule	If m and n are natural numbers, and a is a real number, then a m x a n = a m + n
Example	Rewrite 4 24 3 using a single base and exponent. The product rule states that a m x a n = a m + n Applying the rule to the expression and simplifying we get 4 2 x 4 3 = 4 2 + 3 = 4 5 4 2 = 4 x 4 and 4 3 = 4 x 4 x 4
Think about it this way so	4 2 x 4 3 = ( 4 x 4 ) x ( 4 x 4 x 4 ) = 4 5

Zero as an exponent

According to the zero exponent rule, any nonzero number raised to the power 0 equals 1.

According to the zero exponent rule, any nonzero number raised to the power 0 equals 1.

This means that a ⁰ = 1

Let us understand it using an example.

6 ⁰ = 1

( – 7 ) ⁰ = 1

The above rule can be summarised as –

Zero Rule	If a is any nonzero number, then a 0 = 1
Example	Evaluate the expression ( – 2 ) 0 The zero exponent rule states that a 0 = 1 Applying the rule to the expression we get ( – 2 ) 0 = 1
Think about it this way so	( – 2 ) x 0 number of times = 0

One as an exponent

According to the one exponent rule, any nonzero number raised to the power 1 equals the number itself.

This means that a ¹ = a

Let us understand it using an example.

6 ¹ = 6

( – 7 ) ¹ = – 7

The above rule can be summarised as –

One exponent Rule	If a is any nonzero number, then a 1 = a
Example	Evaluate the expression ( – 2 ) 1 The one exponent rule states that a 1 = a Applying the rule to the expression we get ( – 2 ) 1 =  = -2
Think about it this way so	( – 2 ) x 1 number of times =  – 2

Quotient rule for exponents

We make use of quotient rule when dividing exponents with the same base. The quotient rule for exponents states that if we divide exponents with the same base, then we can subtract the exponents and keep the base unchanged.

This means that if a is any real number and m and n are positive integers, then $\frac{a^m}{a^n} = a^{m-n}$

Let us check the proof of this rule.

We shall divide the proof into three parts, ( i ) when m > n ( ii ) when m = n and ( iii ) when m < n. Let us discuss them one by one.

Case 1 When m > n. In this case, we will have

$\frac{a^m}{a^n} = \frac{a x a x a x a x ax . . . . . . . . . . . . . . . to\: m\: factors}{a x a x a x a x a x a x . . . . . . . . . . . . . . to\: m\: factors}$

⇒ $\frac{a^m}{a^n}$ = a x a x a x a x  . . . . . . . . . . . . . . ( m – n )  factors

⇒ $\frac{a^m}{a^n} = a^{m-n}$  . . . . . . . . . . . . . . . . [ Cancelling n factors in numerators and the denominator ) 

Case 2 When m = n. In this case, we will have,

$\frac{a^m}{a^n} = \frac{a x a x a x a x ax . . . . . . . . . . . . . . . to\: m\: factors}{a x a x a x a x a x a x . . . . . . . . . . . . . . to\: m\: factors}$

= 1 . . . . . . . . . . . ……. . . . . [ Cancelling common factors in numerators and the denominator )

= a ⁰  . . . . . . . . . . . . . . . . [ a ⁰ = 1 ( By definition ) ]

= a^m-m . . . . . . . . . . . . . . . .  [ m – m = 0 ]

= a^m-n . . . . . . . . . . . . . . . .  [ m = n ]

Hence, $\frac{a^m}{a^n} = a^{m-n}$

Case 3 When m < n. In this case, we have,

$\frac{a^m}{a^n} = \frac{a x a x a x a x ax . . . . . . . . . . . . . . . to\: m\: factors}{a x a x a x a x a x a x . . . . . . . . . . . . . . to\: m\: factors}$

⇒ $\frac{a^m}{a^n} = \frac{1}{a x a x a x a x a x a x . . . . . . . . . . . . . . to\: ( n-m )\: factors}$ . . . . . . . .. . . . . . . . . . . .. . . . [ Cancelling common factors in numerators and the denominator )

⇒ $\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$

⇒ $\frac{a^m}{a^n}$ = a^{-( n-m )} 

⇒ $\frac{a^m}{a^n}$ = a^m-n

Hence, $\frac{a^m}{a^n} = a^{m-n}$

Let us understand it using an example

$\frac{5^8}{5^3} = 5^{8-3} = 5^5$

Product of Exponents

If a is any real number and m and n are positive integers, then ( a ^m ) ⁿ = a ^{m n}

Let us understand its proof.

Using the definition, we have,

( a ^m ) ⁿ = a ^m x  a ^m   a ^m x a ^m x  a ^m . . . . . . . . . . . . . . . to n factors

⇒( a ^m ) ⁿ = ( a x a x a x a x . . . . . .  m factors ) x ( a x a x a x a x  . . . . . . . . . . . . to m factors ) x ( a x a x a x a x  . . . . . . . . . . . . to m factors ) . . . . . . . . . . . . . . to n factors

⇒( a ^m ) ⁿ = a x a x a x a x a x . . . . . . . . . . . to ( m n ) factors = a ^{m n}

Hence, ( a ^m ) ⁿ = a ^{m n}

Let us understand it using an example

( 5 ³ ) ² = 5 ^{3 x 2} = 56

Quotient to a Power

The quotient to a power rule states that exponents involving a quotient is equal to the quotients of two exponents. This means that –  a ⁿ / b ⁿ = ( a / b ) ⁿ 

In other words, when we divide one exponent by another, we can simplify it by writing both expressions under the same power and then simplify them in order to get the result. There are however, some conditions that need to be considered during the application of quotient to a power rule in exponents – 

1. Each term has the same index. The index is the superscript number to the left of the exponent, which indicates the degree of the exponent.
   
   For instance, 4 ² = a x 4 = 16
   
   Here the index is 2 as it is indicating the square of 4. 

2. The denominator of the fraction is not zero.

Let us understand it by an example.

10 ³ / 5 ³ = ( 10 / 5 )  ³ = 2 ³

Product to a Power

The product to a Power rule states that exponents involving a product are equal to the product of two exponents. This means that –  a ⁿ b ⁿ = ( a b ) ⁿ

Let us understand it by an example.

2 ³ x 5 ³ = ( 2 x 5 )  ³ = 10 ³

In other words, when we divide one exponent by another, we can simplify it by writing both expressions under the same power and then simplify them in order to get the result.

Use of Rules of Exponents to Perform Operations on Radicals

The rules of exponents that have learnt above are not just used for the purpose of performing operations on exponents. They find their use in solving radical expressions as well. Before we learn more about these, let us recall what we mean by radicals.

What are radicals?

We know that a radical in mathematics is the opposite of an exponent. Let us recall that for any real number “ a”  and a positive integer “ n”, we define a ⁿ  as

a ⁿ =  a x a x a x a x a …… ( n times ).

Here a ⁿ is called the nth power of a. the real number a is called the base and n is called the exponent of the nth power of a.

Radicals on the other hand are same as the root of a number. In other words, a radical is an expression that involves a root, usually a square root or cube root. The root of a number is expressed using the symbol √, for example, √5, ∛10. etc. The horizontal line covering the number is called the vinculum and the number under it is called the radicand. So, if we have a radical, say ⁿ√x, then it is read as “ x radical n ”. Here, “ n “ is the index and “ x “ is the radicand.

Similarity between Rules of Radicals and rules of exponents

In order to observe the similarity between the rules of radicals and the rules of exponents let us first learn about the rules of radicals.

The following are the general rules of radicals – 

If a and b be positive real numbers, then,

1. ( √a )  ² = a
2. ( √a )  ( √b )  = ( √ab )  
3. $\frac{( √a )}{ ( √b )}  = √\frac{a}{b}$
4. (√a + √b ) (√a – √b ) = a – b
5. ( a + √b ) ( a – √b ) = a ² –  b
6. (√a ± √b ) ² = a ± 2 √ab + b
7. (√a + √b ) (√c + √d ) = √ac + √ad + √bc + √bd

Some other rules of radicals that are important are – 

1. If two or more radicals are multiplied with the same index, you can take the radical once and multiply the numbers inside the radicals. 
   This means that ⁿ √ a x ⁿ √ b = ⁿ √ ( a x b )
2. If two radicals are in division with the same index, you can take the radical once and divide the numbers inside the radicals. This means that ⁿ √ a ÷ ⁿ √ b = ⁿ √ ( a ÷  b )
3. One number can be taken out of a square root for every two same numbers multiplied inside the square root. And also, one number can be taken out of a cube root for every three same numbers multiplied inside the cube root and so on. For example, √9 = $\sqrt{3 x 3}$ = 3. Similarly, ∛27 = $\sqrt[3]{3 x 3 x 3}$ = 3
4. A radical with index n can be written as exponent 1/n. This means that ⁿ√a = a ^1/n.

From above we can observe the similarity between the rules of exponents and the rules of radicals. In fact it is the rules of exponents that form the base for the rules of radicals.

Key Facts and Summary

1. For any real number “ a”  and a positive integer “ n”, we define a ⁿ  as a ⁿ =  a x a x a x a x a …… ( n times ). Here a ⁿ is called the nth power of a. the real number a is called the base and n is called the exponent of the nth power of a.
2. a ^{– n} = $\frac{1}{a^n}$
3. a ^m x a  ⁿ= a ^{m + n}
4. According to the zero exponent rule, any nonzero number raised to the power 0 equals 1. This means that a ⁰ = 1
5. According to the one exponent rule, any nonzero number raised to the power 1 equals the number itself. This means that a ¹ = a
6. If a is any real number and m and n are positive integers, then $\frac{a^m}{a^n}$ = a^m-n
7. If a is any real number and m and n are positive integers, then ( ( a ^m ) ⁿ = a ^{m n}
8. ( √a )  ² = a
9. ( √a )  ( √b )  = ( √ab )  
10. $\frac{( √a )}{( √b )}  = √\frac{a}{b}$
11. (√a + √b ) (√a – √b ) = a – b
12. ( a + √b ) ( a – √b ) = a ² –  b
13. (√a ± √b ) ² = a ± 2 √ab + b
14. (√a + √b ) (√c + √d ) = √ac + √ad + √bc + √bd
15. The quotient to a power rule states that exponents involving a quotient are equal to the quotients of two exponents. This means that –  a ⁿ / b ⁿ = ( a / b ) ⁿ 
16. The product to a Power rule states that exponents involving a product are equal to the product of two exponents. This means that –  a ⁿ b ⁿ = ( a b ) ⁿ
17. The rules of exponents that form the base for the rules of radicals.

Recommended Worksheets

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