Easy Way to Remember 0 Exponet Rule

In Mathematics, in that location are unlike laws of exponents. All the rules of exponents are used to solve many mathematical problems which involve repeated multiplication processes. The laws of exponents simplify the multiplication and division operations and aid to solve the issues easily. In this article, we are going to discuss the six important laws of exponents with many solved examples.

Tabular array of Contents:

• Exponent Definition
• Laws of Exponents
  • Powers with Same Base
  • Quotient with Same Base
  • Power of a Ability
  • Product to a Power
  • Quotient to a Power
  • Zip Power Rule
  • Negative Exponent Rule
  • Partial Exponent Rule
• Practice Issues
• FAQs

What are Exponents?

Exponents are used to show repeated multiplication of a number by itself. For example, 7 × 7 × vii tin can be represented as seven³ . Hither, the exponent is 'iii' which stands for the number of times the number 7 is multiplied. 7 is the base here which is the bodily number that is getting multiplied. So basically exponents or powers denotes the number of times a number can be multiplied. If the ability is ii, that means the base of operations number is multiplied two times with itself. Some of the examples are:

• 3 ⁴ = 3×iii×iii×three
• 10 ⁵ = x×10×10×10×ten
• sixteen ³ = 16 × 16 × 16

Suppose, a number 'a' is multiplied by itself due north-times, then it is represented every bit a ^{due north}  where a is the base and n is the exponent.

Exponents follow certain rules that aid in simplifying expressions which are also called its laws. Let united states hash out the laws of exponents in detail.

Rules of Exponents With Examples

As discussed earlier, there are dissimilar   laws or rules defined for exponents. The important laws of exponents are given beneath:

• a ^m ×a ⁿ = a ^g+north
• a ^m /a ^{due north} = a ^{chiliad-due north}
• (a ^m ) ⁿ = a ^mn
• a ^{due north} /b ^northward = (a/b) ⁿ
• a ⁰ = i
• a ^-m = 1/a ^m

• \(\brainstorm{array}{l}a^{\frac{1}{northward}}=\sqrt[northward]{a}\end{assortment} \)

Also, read:At present allow united states hash out all the laws one past i with examples here.

• An Introduction To Exponents
• Exponents And Powers for Class 7
• Exponents And Powers For Class 8

Product With the Same Bases

As per this law, for whatsoever non-zero term a,

• a ^k ×a ⁿ = a ^{chiliad+northward}

where grand and northward are real numbers.

Example 1: What is the simplification of 5^v × 5¹?

Solution: v⁵ × 5¹= 5^5+one  = 5 ⁶

Case 2: What is the simplification of (−6) ^-iv × (−half dozen) ^-7 ?

Solution: (−6) ^-four × (−6) ^-seven = (-6) ^-4-7 = (-6) ^-11

Note: We tin can land that the law is applicable for negative terms likewise. Therefore the term m and n can be any integer.

Quotient with Same Bases

As per this rule,

• a ^m /a ^{due north} = a ^{chiliad-north}

where a is a not-null term and m and n are integers.

Example 3: Find the value when 10^-5 is divided by 10 ^-iii .

Solution: Every bit per the question;

10^-five /x ^-3

= 10 ^-5-(-)3

= x ^-5+3

= 10 ^-2

= i/100

Power Raised to a Power

According to this police force, if 'a' is the base, then the power raised to the power of base 'a' gives the product of the powers raised to the base 'a', such as;

• (a ^m ) ⁿ = a ^mn

where a is a not-nada term and m and due north are integers.

Instance iv: Express viii^three as a power with base ii.

Solution: We have, 2×2×ii = 8 = 2 ^three

Therefore, eight ³ = (2³)³  = 2 ⁹

Product to a Power

As per this rule, for two or more dissimilar bases, if the ability is aforementioned, then;

• a ⁿ b ⁿ = (ab) ⁿ

where a is a non-zero term and north is the integer.

Example 5: Simplify and write the exponential class of: one/8 x five ^-3

Solution: We can write, i/8 = two ^-3

Therefore, two^-3 x five^-3  = (2 × 5)^-3  = x^-3

Quotient to a Power

As per this law, the fraction of 2 different bases with the same power is represented as;

•  a ⁿ /b ⁿ = (a/b) ⁿ

where a and b are not-aught terms and due north is an integer.

Example 6: Simplify the expression and observe the value:15 ³ /5 ³

Solution: We can write the given expression equally;

(15/5) ³ = 3 ³  = 27

Zero Power

Co-ordinate to this dominion, when the power of whatsoever integer is null, then its value is equal to 1, such every bit;

a ⁰ = 1

where 'a' is any non-aught term.

Case vii: What is the value of 5⁰ + 2ⁱⁱ + 4⁰ + 7^ane – 3^one  ?

Solution: 5⁰ + iiⁱⁱ + 4⁰ + vii¹ – 3¹  = 1+4+ane+7-iii= 10

Negative Exponent Rule

Co-ordinate to this rule, if the exponent is negative, we can change the exponent into positive by writing the same value in the denominator and the numerator holds the value 1.

The negative exponent rule is given as:

a^-m = 1/a^k

Example 8:

Find the value of 2 ^-2

Solution:

Here, the exponent is a negative value (i.due east., -2)

Thus, 2 ^-2 tin can be written as 1/2 ²

2 ^-two = 1/2 ^two

2 ^-ii = 1/iv

In other words, we tin say that, if "a" is a non-zero number or non-nada rational number, we can say that a ^-m is the reciprocal of a ^k .

Partial Exponent Rule

The partial exponent rule is used, if the exponent is in the fractional form. The fractional exponent rule is given by:

\(\begin{assortment}{50}a^{\frac{1}{n}}=\sqrt[n]{a}\end{assortment} \)

Here, a is called the base, and 1/north is the exponent, which is in the fractional form. Thus, a ^{1/due north} is said to exist the n th root of a.

Case 9:

Simplify: four ^one/2

Solution:

Here, the exponent is in fractional form. (i.due east., ½)

According to the fractional exponent dominion, four ^1/2 can be written as √iv

(i.due east.,) iv ^1/2 = √4

4 ^1/ii = 2 (As, the square root of 4 is 2)

Hence, the simplified course of four ^ane/2 is 2.

Practice Bug on Laws of Exponents

Simplify the following expressions using the laws of exponents:

1. (4 ² ) ³
2. 4 ⁱⁱ ×4 ^vii
3. 3 ^-3
4. 64 ^i/2
5. 7⁰×ii³

Frequently Asked Questions on Laws of Exponents

What are exponents?

The exponents, also called powers, define how many times we take to multiply the base number. For example, the number two has to be multiplied 3 times and is represented by two³.

What are the different laws of exponents?

The unlike Laws of exponents are:

• a ^thousand ×a ⁿ = a ^m+n
• a ^m /a ⁿ = a ^m-northward
• (a ^chiliad ) ⁿ = a ^mn
• a ^{due north} /b ^north = (a/b) ⁿ
• a ⁰ = 1
• a ^-chiliad = 1/a ^m

What is Power of a power dominion?

In the power of a power rule, nosotros have to multiply the exponent values. For example, (2³)² tin can be written as two^six.

Explicate the zero power rule.

According to the cypher power rule, if the exponent is nada, the event is 1, whatever the base value is. It means that anything raised to the power of 0 is 1. For example, 5⁰ is 1.

Simplify the expression 2².two⁵

In expression 2².2⁵, the base values are the aforementioned, So we tin can add the exponents.
Hence, twoⁱⁱ.ii⁵ = 2²⁺⁵
twoⁱⁱ.2⁵ = two^vii.

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