what is 2 to the negative 2 power

Negative Exponents

Negative exponents tell us that the power of a number is negative and it applies to the reciprocal of the number. We know that an exponent refers to the number of times a number is multiplied by itself. For example, 3ⁱⁱ = three × 3. In the case of positive exponents, we hands multiply the number (base of operations) by itself, but what happens when we accept negative numbers as exponents? A negative exponent is divers equally the multiplicative inverse of the base, raised to the power which is contrary to the given power. In simple words, we write the reciprocal of the number and then solve it like positive exponents. For example, (2/iii)^-2 tin be written every bit (iii/2)².

1.	What are Negative Exponents?
2.	Negative Exponent Rules
3.	Why are Negative Exponents Fractions?
iv.	Multiplying Negative Exponents
5.	How to Solve Negative Exponents?
6.	FAQs on Negative Exponents

What are Negative Exponents?

Nosotros know that the exponent of a number tells united states of america how many times nosotros should multiply the base. For instance, consider 8², viii is the base, and ii is the exponent. Nosotros know that 8² = 8 × 8. A negative exponent tells us, how many times nosotros have to multiply the reciprocal of the base. Consider the 8^-ii, here, the base is 8 and we have a negative exponent (-two). 8^-2 is expressed as i/8² = ane/8×1/8.

Numbers and Expressions with Negative Exponents

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

Negative Exponent	Outcome
two-1	1/2
3-2	1/threeii=1/9
x-iii	1/x3
(2 + 4x)-2	1/(ii+4x)2
(xtwo+ y2)-three	1/(x2+yii)3

Negative Exponent Rules

Nosotros take a ready of rules or laws for negative exponents which make the process of simplification like shooting fish in a barrel. Given beneath are the basic rules for solving negative exponents.

• Rule 1: The negative exponent rule states that for every number 'a' with the negative exponent -northward, take the reciprocal of the base and multiply information technology according to the value of the exponent: a^(-northward)=1/a^{due north}=ane/a×i/a×....due north times
• Rule 2: The rule for a negative exponent in the denominator suggests that for every number 'a' in the denominator and its negative exponent -n, the result can be written as: i/a^{(-due north)}=aⁿ=a×a×....northward times

Permit the states apply these rules and see how they piece of work with numbers.

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

Solution:

• Use the negative exponent rule a^-northward=1/aⁿ
• 2^-ii + 3^-2 = 1/2² + one/three^two = 1/4 + 1/ix
• Take the To the lowest degree Common Multiple (LCM): (9+4)/36 = 13/36

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

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

Solution:

• Use the second rule with a negative exponent in the denominator: ane/a^-n =a^{due north}
• 1/4^-two + 1/ii^-three = 4² + 2³ =xvi + viii = 24

Therefore, 1/four^-2 + 1/2^-three = 24.

Why are Negative Exponents Fractions?

A negative exponent takes u.s.a. to the inverse of the number. In other words, a^-n = ane/aⁿ and 5^-three becomes 1/5ⁱⁱⁱ = one/125. This is how negative exponents modify the numbers to fractions. Allow us take some other example to see how negative exponents modify to fractions.

Example: Solve ii^-one + 4^-two

Solution:

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

Multiplying Negative Exponents

Multiplication of negative exponents is the aforementioned as the multiplication of any other number. Every bit we accept already discussed that negative exponents can be expressed as fractions, so they can easily exist solved later they are converted to fractions. Afterwards this conversion, we multiply negative exponents using the same rules that we utilize for multiplying positive exponents. Let'south understand the multiplication of negative exponents with the post-obit example.

Example: Solve: (4/v)^-3 × (x/three)^-2

• The start pace is to write the expression in its reciprocal class, which changes the negative exponent to a positive 1: (5/4)3×(3/10)two
• Now open the brackets: \(\frac{five^{3} \times 3^{ii}}{iv^{3} \times 10^{2}}\)(∵10^two=(five×two)² =5ⁱⁱ×2²)
• Check the common base and simplify: \(\frac{5^{3} \times 3^{2} \times v^{-2}}{four^{3} \times 2^{2}}\)
• \(\frac{v \times iii^{two}}{iv^{three} \times 4}\)
• 45/four^iv = 45/256

How to Solve Negative Exponents?

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

Example: Solve: (7³) × (iii^-four/21^-2)

Solution:

First, nosotros convert all the negative exponents to positive exponents and so simplify

• Given: \(\frac{7^{3} \times 3^{-4}}{21^{-2}}\)
• Convert the negative exponents to positive past writing the reciprocal of the detail number:\(\frac{7^{three} \times 21^{2}}{3^{4}}\)
• Utilize the rule: (ab)^{due north} = a^north × bⁿ and dissever the required number (21).
• \(\frac{seven^{3} \times 7^{ii} \times 3^{2}}{3^{four}}\)
• Utilize the rule: a^m × aⁿ = a^(m+n) to combine the common base (7).
• seven⁵/3² =16807/nine

Important Notes:

Notation the post-obit points which should exist remembered while we work with negative exponents.

• Exponent or ability means the number of times the base needs to be multiplied past itself.
  a^m = a × a × a ….. m times
  a^{-one thousand} = i/a × one/a × ane/a ….. m times
• a^-n is besides known every bit the multiplicative inverse of a^north.
• If a^-m = a^-northward so m = north.
• The relation betwixt the exponent (positive powers) and the negative exponent (negative power) is expressed equally a^x=1/a^-x

Topics Related to Negative Exponents

Check the given manufactures similar or related to the negative exponents.

• Exponent Rules
• Exponents
• Multiplying Exponents
• Fractional Exponents
• Irrational Exponents
• Exponents Formula
• Exponential Equations

Examples of Negative Exponents

1. Example one: Observe the solution of the given problem (3² + 4²)^-2 by using negative exponents rules.

   Solution:

   (3² + 4ⁱⁱ)^-two = (9 + 16)^-2 = (25)^-ii = 1/25² = 1/625. Therefore, (3² + 4ⁱⁱ)^-2 = ane/625

2. Example 2: Notice the value of ten in 27/three^-x = 3⁶

   Solution:

   Hither we have negative exponents with variables.

   27/3^-10 = 3⁶
   three³/three^-10 = 3^half-dozen
   3³ × 3^x = 3⁶
   3^(three+x) = 3^{half dozen}

   If bases are aforementioned then exponents must be equal, and then, ten + 3 = 6, x = iii. Therefore, the value of x = three.

3. Instance 3: Simplify the following using negative exponent rules: (two/iii)^-2 + (five)^-i

   Solution: By using negative exponent rules, we tin write (2/iii)^-2 as (three/2)ⁱⁱ and (5)^-one as 1/5. So, nosotros tin can simplify the given expression as,
   (3/2)^two + ane/v
   9/4 + 1/5
   (45+4)/20
   49/xx
   Therefore, (2/iii)^-2 + (v)^-1 is simplified to 49/20.

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Practice Questions on Negative Exponents

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FAQs on Negative Exponents

What are Negative Exponents?

When nosotros have negative numbers as exponents, we phone call them negative exponents. For instance, in the number 2^-8, -8 is the negative exponent of base 2.

Do negative exponents make negative numbers?

This is not truthful that negative exponents give negative numbers. Being positive or negative depends on the base of the number. Negative numbers give a negative result when their exponent is odd and they give a positive result when the exponent is even. For example, (-five)³ = -125, (-five)⁴ = 625. A positive number with a negative exponent will ever requite a positive number. For example, two^-three = 1/8, which is a positive number.

How to Simplify Negative Exponents?

Negative exponents are simplified using the same laws of exponents that are used to solve positive exponents. For example, to solve: three^-3 + 1/2^-4, first we modify these to their reciprocal form: 1/iii³ + two⁴, then simplify ane/27 + sixteen. Taking the LCM, [i+ (sixteen × 27)]/27 = 433/27.

What is the Rule for Negative Exponents?

There are 2 master negative exponent rules that are given below:

• Let a be the base of operations and n be the exponent, we have, a^-n = one/aⁿ.
• 1/a^-n = aⁿ

How to Split up Negative Exponents?

Dividing exponents with the aforementioned base is the same as multiplying exponents, but get-go, we need to convert them to positive exponents. We know that when the exponents with the same base are multiplied, the powers are added and we apply the same rule while dividing exponents. For example, to solve y⁵ ÷ y^-iii, or, y^v/y^-3, offset we change the negative exponent (y^-3) to a positive ane by writing its reciprocal. This makes it: y⁵ × y³ = y^(5+three) = y^viii.

How to Multiply Negative Exponents?

While multiplying negative exponents, showtime we demand to convert them to positive exponents past writing the respective numbers in their reciprocal form. In one case they are converted to positive ones, we multiply them using the aforementioned rules that we utilise for multiplying positive exponents. For instance, y^-5 × y^-2 = 1/y⁵ × 1/y^two = 1/y⁽⁵⁺ⁱⁱ⁾ = 1/y⁷.

Why are Negative Exponents Reciprocals?

When nosotros need to change a negative exponent to a positive 1, we are supposed to write the reciprocal of the given number. Then, the negative sign on an exponent indirectly means the reciprocal of the given number, in the same style equally a positive exponent ways the repeated multiplication of the base.

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents can be solved past taking the reciprocal of the fraction. And then, find the value of the number past taking the positive value of the given negative exponent. For example, (3/four)^-two can be solved by taking the reciprocal of the fraction, which is 4/3. Now, observe the positive exponent value of 4/3, which is (four/3)² = 4^two/3^two. This results in sixteen/9 which is the final answer.

What is 10 to the Negative Power of 2?

10 to the negative power of two is represented as 10^-2, which is equal to (ane/10²) = 1/100.

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Source: https://www.cuemath.com/algebra/negative-exponents/

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