Exponents Powers: Formulas, Laws & Solved Examples

Exponents and powers are ways to represent large numbers in simplified terms or standard form. The factor multiplied by itself will be the base and the number of times the factor is multiplied will be the exponent. For instance, to show 5×5×5×5×5×5 in a simplified way, we can write it as 5⁶. Here, 5 is the base and 6 is the exponent and the whole expression, 5⁶ will be known as power. In simple terms, power is an expression that shows repeated multiplication of the same number. 

Key terms: Exponent, Power, an, integers, base, Addition, Subtraction, Division, Multiplication, Negative exponent law

Read Also: Rationalize the Denominator

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What is an exponent?

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An exponent is a number that represents the number of times a factor is multiplied by itself. If number 9 is multiplied n number of times, then it will be represented as:

9×9×9×............n= 9ⁿ

Thus, 9ⁿ will be known as 9 raised to power n. Thus, exponents are also sometimes called indices or power. 

Examples:

3 × 3 × 3 = 3³

5 × 5 × 5 × 5 = 5⁴

6 × 6 = 6²

Read Also: Greatest Integer Function

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Formula

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An exponent tells us the number of times a factor must be multiplied by itself to get the desired result. Thus, number 'a' raised to power 'n' can be represented as:

aⁿ = a × a × a × .........… × a

Here, a and n are any numbers 

an is also known as the nth power of a, where a is the base and n is the exponent or power. 'a' is multiplied 'n' number of times, making exponentiation a shorthand method of repeated multiplication.

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Laws of exponent

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Laws of exponents are based upon the powers they carry.

• Bases, multiplying the like ones, add the exponents while keeping the base the same. This is known as the multiplication law.
• Bases raised with power to another, multiply the exponents while keeping the base the same.
• Bases dividing like ones, subtract the denominator from the numerator exponent while keeping the base the same. This is known as the division law.

If 'a' is any number and 'm', 'n' are positive integers denoting the power, then;

Multiplication law

According to the multiplication law, the product of two exponents having the same base but different powers is equal to the base raised to the sum of two powers.

a^m × aⁿ = a^m+n

Negative exponent law

A base with negative power equals the reciprocal of it with positive power.

a^-m = 1/am

Read Also: Distance Formula and Derivation of Coordinate Geometry

Division law

According to this law, when two exponents with the same bases but different powers are divided, it results in the base being raised to the difference between two powers.

a^m ÷ aⁿ = a^m-n

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Rules of exponent

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Rules of exponents are followed based upon the laws. 

If ‘a’ & ‘b’ are the integers and ‘m’ & ‘n’ are the values for powers, then the rules for exponents and powers will be;

i) a⁰ = 1

According to this rule, if the power of any integer is zero, then the output will be equal to one.

Example: 6⁰=1

ii) (a^m)ⁿ = a(^mn)

‘a’ raised to the power ‘m’ raised to the power ‘n’ will be equal to ‘a’ raised to the power product of ‘m’, ‘n’.

Example: (5²)³ = 5^{2 x 3}

iii) a^m× b^m =(ab)^m

Multiplying ‘a’ raised to the power ‘m’ and ‘b’ raised to the power ‘m’ will result in the product of ‘a’ and ‘b’ whole raised to the power ‘m’.

Example: 4² × 7² =(4×7)²

iv) a^m/b^m= (a/b)^m

Dividing ‘a’ raised to the power ‘m’ and ‘b’ raised to the power ‘m’ will result in the division of ‘a’ multiplied by ‘b’ whole raised to the power ‘m’.

Example: 2²/6² = (2/6)²

Also Read: Real-Valued Functions

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Things to remember

• Exponent is a number that represents the number of times a factor is multiplied by itself.
• Formula for exponent: aⁿ = a × a × a × .........… × a
• Multiplication law- a^m × aⁿ = a^{m + n}
• Division law- a^m ÷ aⁿ = a^m-n

Read Also: Section Formula in Coordinate Geometry