Negative exponents tell united state that the strength of a number is negative and it uses to the mutual of the number. We understand that an exponent describes the variety of times a number is multiplied by itself. Because that example, 32 = 3 × 3. In the instance of hopeful exponents, we easily multiply the number (base) by itself, however what happens as soon as we have an adverse numbers together exponents? A negative exponent is characterized as the multiplicative inverse of the base, raised to the power which is opposite come the given power. In basic words, we compose the reciprocal of the number and also then resolve it like optimistic exponents. For example, (2/3)-2 have the right to be composed as (3/2)2.

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

What are an adverse Exponents?


We know that the exponent the a number tells us how many times we should multiply the base. For example, consider 82, 8 is the base, and 2 is the exponent. We know that 82 = 8 × 8. A negative exponent speak us, how countless times we need to multiply the reciprocal of the base. Think about the 8-2, here, the basic is 8 and we have a an adverse exponent (-2). 8-2 is expressed together 1/82 = 1/8×1/8.

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Numbers and Expressions with an adverse Exponents

Here are a couple of examples i beg your pardon express an adverse exponents with variables and also numbers. Watch the table to see how the number is created in the reciprocal kind 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 a set of rule or legislations for an unfavorable exponents which make the process of leveling easy. Given below are the an easy rules because that solving an adverse exponents.

Rule 1: The an adverse exponent dominance states that for every number 'a' v the negative exponent -n, take it the mutual of the base and also multiply it according to the worth of the exponent: a(-n)=1/an=1/a×1/a×....n timesRule 2: The rule for a an unfavorable exponent in the denominator suggests that for every number 'a' in the denominator and its negative exponent -n, the an outcome can be written as: 1/a(-n)=an=a×a×....n times

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Let us use these rules and see exactly how they work with numbers.

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

Solution:

Use the an unfavorable exponent preeminence 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 2nd rule v 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 station of the number. In other words, a-n = 1/an and 5-3 becomes 1/53 = 1/125. This is how negative exponents change the number to fractions. Let united state take another example to view how an adverse exponents change to fractions.

Example: solve 2-1 + 4-2

Solution:

2-1 have the right to be created as 1/2 and also 4-2 is created as 1/42. Therefore, an unfavorable exponents get changed to fractions as soon as the authorize of their exponent changes.


Multiplying negative Exponents


Multiplication of negative exponents is the same as the multiplication of any kind of other number. Together we have currently discussed that an adverse exponents can be expressed as fractions, for this reason they can conveniently be resolved after they room converted to fractions. After this conversion, we multiply an adverse exponents making use of the exact same rules the we apply for multiplying confident exponents. Let's know the multiplication of an adverse exponents through the complying with example.

Example: Solve: (4/5)-3 × (10/3)-2

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

How to Solve an unfavorable Exponents?


Solving any equation or expression is all around operating on those equations or expressions. Similarly, solving an unfavorable exponents is around the leveling of terms with negative exponents and then applying the given arithmetic operations.

Solution:

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

Given: (frac7^3 imes 3^-421^-2)Convert the an adverse exponents to confident by creating the mutual of the certain number:(frac7^3 imes 21^23^4)Use the rule: (ab)n = an × bn and also split the required number (21).(frac7^3 imes 7^2 imes 3^23^4)Use the rule: to be × an = a(m+n) to integrate the typical base (7).75/32 =16807/9

Important Notes:

Note the complying with points which need to be remembered when we occupational with an unfavorable exponents.

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

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