Negative index number tell us that the power of a number is an adverse and it uses to the reciprocal of the number. We recognize that an exponent describes the variety of times a number is multiply by itself. Because that example, 32 = 3 × 3. In the instance of positive exponents, we quickly multiply the number (base) through itself, however what happens once we have an adverse numbers together exponents? A an unfavorable exponent is characterized as the multiplicative station of the base, elevated to the power which is opposite come the provided power. In basic words, we compose the reciprocal of the number and then solve it like confident exponents. Because that example, (2/3)-2 can be composed as (3/2)2.

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1.What are an adverse Exponents?
2.Negative Exponent Rules
3.Why are negative Exponents Fractions?
4.Multiplying an unfavorable Exponents
5. How come Solve an unfavorable Exponents?
6. FAQs on negative Exponents

What are negative Exponents?


We recognize that the exponent of a number tells us how many times we must multiply the base. Because that example, consider 82, 8 is the base, and 2 is the exponent. We recognize that 82 = 8 × 8. A an unfavorable exponent tells us, how countless times we have to multiply the reciprocal of the base. Take into consideration the 8-2, here, the basic is 8 and also we have a an adverse exponent (-2). 8-2 is expressed as 1/82 = 1/8×1/8.

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Numbers and also Expressions with an adverse Exponents

Here are a couple of examples i m sorry express an adverse exponents with variables and numbers. Observe the table to see how the number is written in that reciprocal form and just how the authorize of the powers changes.

Negative ExponentResult
2-11/2
3-21/32=1/9
x-31/x3
(2 + 4x)-21/(2+4x)2
(x2+ y2)-31/(x2+y2)3

Negative Exponent Rules


We have actually a set of rules or regulations for an unfavorable exponents which make the process of simplification easy. Given below are the an easy rules because that solving negative exponents.

Rule 1: The negative exponent dominion states the for every number 'a' v the an adverse exponent -n, take the mutual of the base and multiply it follow to the value of the exponent: a(-n)=1/an=1/a×1/a×....n timesRule 2: The dominance for a an adverse exponent in the denominator argues that because that every number 'a' in the denominator and also its negative exponent -n, the result can be created as: 1/a(-n)=an=a×a×....n times

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Let us apply these rules and also see exactly how they occupational with numbers.

Example 1: Solve: 2-2 + 3-2

Solution:

Use the an unfavorable exponent ascendancy a-n=1/an2-2 + 3-2 = 1/22 + 1/32 = 1/4 + 1/9

Therefore, 2-2 + 3-2 = 13/36

Example 2: Solve: 1/4-2 + 1/2-3

Solution:

Use the second rule with a negative exponent in the denominator: 1/a-n =an1/4-2 + 1/2-3 = 42 + 23 =16 + 8 = 24

Therefore, 1/4-2 + 1/2-3 = 24.


Why are negative Exponents Fractions?


A an unfavorable exponent takes united state to the inverse of the number. In various other words, a-n = 1/an and also 5-3 i do not care 1/53 = 1/125. This is how an adverse exponents change the numbers to fractions. Let us take one more example to view how negative exponents readjust to fractions.

Example: solve 2-1 + 4-2

Solution:

2-1 deserve to be written as 1/2 and 4-2 is written as 1/42. Therefore, an adverse exponents get changed to fractions when the authorize of your exponent changes.


Multiplying an adverse Exponents


Multiplication of negative exponents is the very same as the multiplication of any other number. As we have already discussed that an adverse exponents can be expressed as fractions, for this reason they can easily be resolved after they are converted come fractions. After this conversion, us multiply an unfavorable exponents using the exact same rules that we use for multiplying positive exponents. Let's recognize the multiplication of an unfavorable exponents through the complying with example.

Example: Solve: (4/5)-3 × (10/3)-2

The an initial step is to compose the expression in its reciprocal form, which transforms the an adverse exponent come a positive one: (5/4)3×(3/10)2Now open the brackets: \(\frac5^3 \times 3^24^3 \times 10^2\)(∵102=(5×2)2 =52×22)Check the usual base and simplify: \(\frac5^3 \times 3^2 \times 5^-24^3 \times 2^2\)\(\frac5 \times 3^24^3 \times 4\)45/44 = 45/256

How to Solve negative Exponents?


Solving any type of equation or expression is all about operating top top those equations or expressions. Similarly, solving negative exponents is about the leveling of state with an unfavorable exponents and also then applying the provided arithmetic operations.

Solution:

First, we convert all the negative exponents to optimistic exponents and also then simplify

Given: \(\frac7^3 \times 3^-421^-2\)Convert the an adverse exponents to confident by creating the reciprocal of the certain number:\(\frac7^3 \times 21^23^4\)Use the rule: (ab)n = one × bn and split the forced number (21).\(\frac7^3 \times 7^2 \times 3^23^4\)Use the rule: to be × an = a(m+n) to combine the common base (7).75/32 =16807/9

Important Notes:

Note the following points which should be remembered if we job-related with negative exponents.

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Exponent or power way the variety of times the base demands to be multiply by itself.am = a × a × a ….. M timesa-m = 1/a × 1/a × 1/a ….. M timesa-n is likewise known as the multiplicative station of an.If a-m = a-n climate m = n.The relation between the exponent (positive powers) and also the an adverse exponent (negative power) is expressed together ax=1/a-x

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