How To Make Exponents Positive

Negative Exponents

A negative exponent is defined equally the multiplicative inverse of the base, raised to the power which is of the reverse sign of the given power. In uncomplicated words, we write the reciprocal of the number and and so solve it like positive exponents. For example, (2/three)^-ii can exist written as (3/2)². We know that an exponent refers to the number of times a number is multiplied by itself. For example, 3^two = 3 × 3. In the case of positive exponents, nosotros easily multiply the number (base) by itself, but in case of negative exponents, nosotros multiply the reciprocal of the number past itself. For example, iii^-ii = 1/iii × 1/iii.

Let us learn more than about negative exponents along with related rules and solve more examples.

1.	What are Negative Exponents?
two.	Numbers and Expressions with Negative Exponents
three.	Negative Exponent Rules
iv.	Why are Negative Exponents Fractions?
5.	Negative Fraction Exponents
6.	Multiplying Negative Exponents
7.	How to Solve Negative Exponents?
8.	FAQs on Negative Exponents

What are Negative Exponents?

Nosotros know that the exponent of a number tells us how many times we should multiply the base. For example, in 8², 8 is the base of operations, and ii is the exponent. Nosotros know that 8^two = 8 × 8. A negative exponent tells us, how many times nosotros have to multiply the reciprocal of the base. Consider the eight^-2, here, the base is 8 and we accept a negative exponent (-2). eight^-two is expressed equally 1/eight × 1/8 = one/8².

Negative Exponents

Numbers and Expressions with Negative Exponents

Here are a few examples which express negative exponents with variables and numbers. Observe the table given below to see how the number/expression with a negative exponent is written in its reciprocal form and how the sign of the powers changes.

Negative Exponent	Event
2^-1	1/two
iii^-2	1/3²= 1/ix
ten^-3	1/tenⁱⁱⁱ
(2 + 4x)^-2	1/(2 + 4x)²
(xⁱⁱ + y²)^-3	one/(x² + yⁱⁱ)³

Negative Exponent Rules

We have a gear up of rules or laws for negative exponents which make the process of simplification easy. Given below are the basic rules for solving negative exponents.

Rule ane: The negative exponent rule states that for a base 'a' with the negative exponent -n, take the reciprocal of the base (which is 1/a) and multiply it by itself due north times.
i.eastward., a^(-n) = 1/a × ane/a × ... n times = 1/aⁿ
Rule 2: The dominion is the same fifty-fifty when at that place is a negative exponent in the denominator.
i.e., 1/a^(-north) = a × a × ... .north times = aⁿ

Negative Exponents Rules

Permit u.s.a. employ these rules and come across how they work with numbers.

Example 1: Solve: ii^-2 + three^-two

Solution:

Use the negative exponent rule a^-n = 1/aⁿ
2^-ii+ 3^-2 = one/2²+ 1/3² = one/4 + i/9
Take the Least Common Multiple (LCM): (9 + four)/36 = 13/36

Therefore, 2^-ii + iii^-ii = 13/36

Case 2: Solve: 1/four^-ii + ane/2^-3

Solution:

Use the second rule with a negative exponent in the denominator: one/a^{-due north}=aⁿ
1/4^-2+ 1/2^-three= 4²+ 2^three =16 + eight = 24

Therefore, i/4^-2 + i/ii^-3 = 24.

Negative Exponents are Fractions

A negative exponent takes united states of america to the inverse of the number. In other words, a^-n = i/aⁿ and 5^-three becomes 1/5³ = one/125. This is how negative exponents change the numbers to fractions. Let u.s. take some other example to see how negative exponents change to fractions.

Instance: Limited 2^-1 and four^-2 as fractions.

Solution:

two^-1 can be written as 1/two and iv^-2 is written equally 1/4ⁱⁱ. Therefore, negative exponents get changed to fractions when the sign of their exponent changes.

Negative Fraction Exponents

Sometimes, we might accept a negative fractional exponent like 4^-3/2. Nosotros can apply the aforementioned rule a^{-due north} = i/aⁿ to express this in terms of a positive exponent. i.e., 4^-3/2 = i/4^3/two. Farther, we tin can simplify this using the exponent rules.

4^-three/2 = 1/4^3/two

= ane / (two²)^3/2

= 1 / two^three

= one/8

Multiplying Negative Exponents

Multiplication of negative exponents is the aforementioned as the multiplication of any other number. As we accept already discussed that negative exponents can be expressed as fractions, so they can easily exist solved after they are converted to fractions. After this conversion, nosotros multiply negative exponents using the aforementioned multiplication rule that nosotros apply for multiplying positive exponents. Let us understand the multiplication of negative exponents with the post-obit instance.

Example: Solve: (iv/5)^-3× (10/3)^-two

The commencement step is to write the expression in its reciprocal class, which changes the negative exponent to a positive one: (5/4)³ × (3/10)ⁱⁱ
At present open up the brackets: \(\frac{5^{3} \times 3^{two}}{4^{3} \times 10^{2}}\)
We know that 10²=(5×2)²=5²×2², and so we can substitute ten²by v²×2ⁱⁱ. So nosotros will check the mutual base and simplify: \(\frac{5^{iii} \times 3^{two} \times 5^{-2}}{iv^{3} \times 2^{2}}\)
\(\frac{5 \times 3^{2}}{4^{3} \times iv}\)
45/four^four= 45/256

How to Solve Negative Exponents?

To solve expressions involving negative exponents, first convert them into positive exponents using one of the following rules and simplify:

a^-north = i/aⁿ
ane/a^-n = aⁿ

Example: Solve: (7³) × (three^-iv/21^-2)

Solution:

First, we convert all the negative exponents to positive exponents then simplify.

Given: \(\frac{7^{3} \times 3^{-4}}{21^{-2}}\)
Convert the negative exponents to positive by applying the higher up rules:\(\frac{7^{three} \times 21^{2}}{iii^{4}}\)
Use the dominion: (ab)ⁿ = aⁿ × b^{due north} and split the required number (21).
\(\frac{7^{three} \times 7^{two} \times 3^{2}}{3^{4}}\)
Use the rule: a^m× aⁿ= a^{(yard+northward)} to combine the common base of operations (7).
vii⁵/3ⁱⁱ =16807/9

Important Notes on Negative Exponents:

Exponent or power ways the number of times the base needs to be multiplied by itself.
a^m = a × a × a ….. m times
a^-m = 1/a × 1/a × 1/a ….. m times
a^-northward is also known as the multiplicative inverse of a^north.
If a^-g = a^-north and then one thousand = n.
The relation betwixt the exponent (positive powers) and the negative exponent (negative power) is expressed every bit a¹⁰=i/a^-x

☛ Related Topics:

Negative Exponents Calculator
Exponent Rules Calculator
Exponent Calculator

Examples of Negative Exponents

Example 1: Discover the solution of the given expression (3² + iv^two)^-2 .

Solution:

The given expression is,

(three² + 4²)^-two = (nine + xvi)^-2
= (25)^-2
= 1/25^two (by negative exponents rule)
= i/625.
Therefore, (3^two + 4^two)^-2 = 1/625

Answer: 1/625
Case two: Find the value of x in 27/3^-ten = three⁶

Solution:

Here we take negative exponents with variables.

27/iii^-x = 3^six
iii³/three^-x= 3^half-dozen
threeⁱⁱⁱ × 3^x= 3⁶
three^{(three + x)}= 3^{half dozen}

If bases are the same and so exponents must be equal, so, 3 + x = 6. Solving this, x = 3.

Reply: x = 3.
Example three: Simplify the post-obit using negative exponent rules: (2/three)^-2 + (5)^-1

Solution:

By using negative exponent rules, we can write (ii/three)^-2 equally (3/2)² and (five)^-1 every bit 1/5. And then, we can simplify the given expression as,
= (3/2)² + i/5
= 9/4 + 1/five
After taking the LCM, nosotros become, (45 + iv)/20
49/20
Therefore, (two/3)^-two + (5)^-1 is simplified to 49/20.

Reply: 49/20.

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FAQs on Negative Exponents

What practise Negative Exponents Hateful?

The negative exponents hateful the negative numbers that are present in place of exponents. For instance, in the number 2^-eight, -8 is the negative exponent of base two.

Do negative exponents Result in Negative Numbers?

No, it is not necessary that negative exponents give negative numbers. For example, two^-3 = i/8, which is a positive number.

How to Calculate Negative Exponents?

Negative exponents are calculated using the aforementioned laws of exponents that are used to solve positive exponents. For instance, to solve: 3^-3 + 1/2^-4, showtime nosotros change these to their reciprocal form: 1/3³ + 2⁴, then simplify ane/27 + 16. Taking the LCM, [one+ (sixteen × 27)]/27 = 433/27.

What is the Dominion for Negative Exponents?

There are 2 main rules that are helpful when dealing with negative exponents:

a^-north = 1/a^north
1/a^-n = aⁿ

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents tin be solved by taking the reciprocal of the fraction. Then, find the value of the number by taking the positive value of the given negative exponent. For example, (three/4)^-2 = (4/3)² = 4^two/3². This results in xvi/9 which is the terminal answer.

How to Carve up Negative Exponents?

Dividing exponents with the same base results in the subtraction of exponents. For instance, to solve y⁵÷ y^-three = y^5-(-3)= y⁸. This can be simplified in an alternative mode besides. i.eastward., y^v÷ y^-3 = y^five/y^-three, first we change the negative exponent (y^-3) to a positive ane by writing its reciprocal. This makes it: y⁵ × y³ = y⁽⁵⁺³⁾ = y⁸.

How to Multiply Negative Exponents?

While multiplying negative exponents, first we demand to convert them to positive exponents by writing the respective numbers in their reciprocal form. Once they are converted to positive ones, we multiply them using the same rules that we apply for multiplying positive exponents. For instance, y^-v × y^-2 = i/y⁵ × 1/yⁱⁱ= 1/y^(five+ii)= 1/y⁷

Why are Negative Exponents Reciprocals?

When we demand to change a negative exponent to a positive one, nosotros are supposed to write the reciprocal of the given number. And so, the negative sign on an exponent indirectly means the reciprocal of the given number, in the aforementioned style equally a positive exponent means the repeated multiplication of the base.

What is 10 to the Negative Power of 2?

x to the negative ability of ii is represented equally 10^-2, which is equal to (i/10ⁱⁱ) = ane/100.