Negative exponents tell united state that the power of a number is an unfavorable and it applies to the mutual of the number. We recognize that one exponent refers to the number of times a number is multiplied by itself. For example, 32 = 3 × 3. In the instance of positive exponents, we quickly multiply the number (base) by itself, however what happens once we have an adverse numbers as exponents? A negative exponent is characterized as the multiplicative station of the base, increased to the strength which is opposite to the provided power. In an easy words, we write the mutual of the number and also then resolve it like confident exponents. Because that example, (2/3)-2 deserve to be written as (3/2)2.
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| 1. | What are negative Exponents? |
| 2. | Negative Exponent Rules |
| 3. | Why are an unfavorable Exponents Fractions? |
| 4. | Multiplying an adverse Exponents |
| 5. | How to Solve negative Exponents? |
| 6. | FAQs on an unfavorable Exponents |
What are negative Exponents?
We know that the exponent the a number tells united state how numerous times we should multiply the base. For example, think about 82, 8 is the base, and also 2 is the exponent. We understand that 82 = 8 × 8. A an adverse exponent tells us, how numerous times we need to multiply the reciprocal of the base. Take into consideration the 8-2, here, the basic is 8 and we have actually a negative exponent (-2). 8-2 is expressed as 1/82 = 1/8×1/8.
Numbers and also Expressions with an adverse Exponents
Here are a few examples i m sorry express negative exponents with variables and also numbers. Watch the table to see just how the number is created in that is reciprocal type and just how the sign of the powers changes.
| 2-1 | 1/2 |
| 3-2 | 1/32=1/9 |
| x-3 | 1/x3 |
| (2 + 4x)-2 | 1/(2+4x)2 |
| (x2+ y2)-3 | 1/(x2+y2)3 |
Negative Exponent Rules
We have a collection of rules or laws for an unfavorable exponents which do the process of simplification easy. Given below are the an easy rules for solving an unfavorable exponents.
Rule 1: The an adverse exponent ascendancy states that for every number 'a' with the negative exponent -n, take it the mutual of the base and multiply it follow to the worth of the exponent: a(-n)=1/an=1/a×1/a×....n timesRule 2: The dominion for a negative exponent in the denominator argues that for every number 'a' in the denominator and also its negative exponent -n, the result can be composed as: 1/a(-n)=an=a×a×....n times
Let us use these rules and see just how they work-related with numbers.
Example 1: Solve: 2-2 + 3-2
Solution:
Use the negative exponent dominion a-n=1/an2-2 + 3-2 = 1/22 + 1/32 = 1/4 + 1/9Therefore, 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 = 24Therefore, 1/4-2 + 1/2-3 = 24.
Why are an adverse Exponents Fractions?
A an adverse exponent takes united state to the train station 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 adjust the number to fractions. Let united state take another example to view how negative exponents change to fractions.
Example: fix 2-1 + 4-2
Solution:
2-1 have the right to be written as 1/2 and 4-2 is written as 1/42. Therefore, an adverse exponents get changed to fractions when the sign of their exponent changes.
Multiplying an adverse Exponents
Multiplication of an unfavorable exponents is the exact same as the multiplication of any kind of other number. As we have currently discussed that negative exponents can be expressed together fractions, for this reason they can easily be addressed after they room converted come fractions. After ~ this conversion, us multiply an adverse exponents utilizing the very same rules the we use for multiplying hopeful exponents. Let's know the multiplication of negative exponents with the complying with example.
Example: Solve: (4/5)-3 × (10/3)-2
The an initial step is to write the expression in its mutual form, which changes the an adverse exponent come a positive one: (5/4)3×(3/10)2Now open 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 = 45/256How come Solve an adverse Exponents?
Solving any kind of equation or expression is all around operating ~ above those equations or expressions. Similarly, solving an unfavorable exponents is around the leveling of terms with an adverse exponents and also then applying the given arithmetic operations.
Solution:
First, we transform all the an unfavorable exponents to optimistic exponents and then simplify
Given: (frac7^3 imes 3^-421^-2)Convert the an unfavorable exponents to hopeful by composing the reciprocal of the certain number:(frac7^3 imes 21^23^4)Use the rule: (ab)n = one × bn and also split the required number (21).(frac7^3 imes 7^2 imes 3^23^4)Use the rule: to be × one = a(m+n) to combine the usual base (7).75/32 =16807/9Important Notes:
Note the complying with points which must be remembered while we occupational with an unfavorable exponents.
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