what do i do if the power has minus 4 next to it

Negative Exponents

Negative exponents tell united states of america that the power of a number is negative and information technology applies to the reciprocal of the number. Nosotros know that an exponent refers to the number of times a number is multiplied past itself. For instance, 3ⁱⁱ = 3 × 3. In the case of positive exponents, we easily multiply the number (base) by itself, but what happens when we have negative numbers as exponents? A negative exponent is divers as the multiplicative inverse of the base of operations, raised to the power which is opposite to the given power. In elementary words, we write the reciprocal of the number and and so solve it like positive exponents. For example, (2/3)^-2 tin be written as (three/2)ⁱⁱ.

one.	What are Negative Exponents?
2.	Negative Exponent Rules
3.	Why are Negative Exponents Fractions?
4.	Multiplying Negative Exponents
5.	How to Solve Negative Exponents?
six.	FAQs on Negative Exponents

What are Negative Exponents?

We know that the exponent of a number tells u.s.a. how many times nosotros should multiply the base. For example, consider 8², 8 is the base, and 2 is the exponent. Nosotros know that 8^two = 8 × eight. A negative exponent tells u.s.a., how many times we take to multiply the reciprocal of the base of operations. Consider the 8^-2, here, the base is 8 and we accept a negative exponent (-2). 8^-ii is expressed as ane/viii² = 1/viii×i/eight.

Numbers and Expressions with Negative Exponents

Here are a few examples which express negative exponents with variables and numbers. Notice the table to run into how the number is written in its reciprocal grade and how the sign of the powers changes.

Negative Exponent	Result
2-1	ane/2
3-two	1/3ii=1/9
x-3	ane/x3
(2 + 4x)-ii	one/(two+4x)2
(xtwo+ y2)-3	1/(x2+yii)3

Negative Exponent Rules

Nosotros accept a fix 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 1: The negative exponent dominion states that for every number 'a' with the negative exponent -n, take the reciprocal of the base of operations and multiply it co-ordinate to the value of the exponent: a^(-n)=1/aⁿ=one/a×1/a×....due north times
• Rule ii: 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: 1/a^(-n)=a^northward=a×a×....n times

Let united states utilise these rules and encounter how they work with numbers.

Example one: Solve: 2^-2 + 3^-2

Solution:

• Use the negative exponent rule a^-n=1/aⁿ
• 2^-ii + 3^-2 = ane/two² + 1/3ⁱⁱ = ane/iv + 1/9
• Take the Least Common Multiple (LCM): (9+4)/36 = 13/36

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

Example ii: Solve: 1/4^-2 + 1/2^-iii

Solution:

• Use the 2d rule with a negative exponent in the denominator: 1/a^-n =aⁿ
• ane/4^-ii + 1/2^-3 = 4² + ii³ =16 + viii = 24

Therefore, ane/4^-2 + 1/two^-iii = 24.

Why are Negative Exponents Fractions?

A negative exponent takes the states to the inverse of the number. In other words, a^-n = 1/aⁿ and 5^-iii becomes i/5³ = 1/125. This is how negative exponents change the numbers to fractions. Allow us take another example to see how negative exponents change to fractions.

Case: Solve 2^-1 + 4^-two

Solution:

2^-1 can exist written as 1/ii and 4^-2 is written as 1/4². Therefore, negative exponents become changed to fractions when the sign of their exponent changes.

Multiplying Negative Exponents

Multiplication of negative exponents is the same as the multiplication of any other number. As nosotros accept already discussed that negative exponents tin be expressed as fractions, then they tin can hands be solved after they are converted to fractions. After this conversion, we multiply negative exponents using the aforementioned rules that we apply for multiplying positive exponents. Let'due south empathize the multiplication of negative exponents with the post-obit instance.

Example: Solve: (4/5)^-three × (10/iii)^-2

• The beginning step is to write the expression in its reciprocal form, which changes the negative exponent to a positive 1: (5/iv)3×(3/10)2
• Now open the brackets: \(\frac{v^{3} \times 3^{2}}{4^{iii} \times 10^{2}}\)(∵10^two=(5×2)^two =5²×2²)
• Bank check the mutual base and simplify: \(\frac{5^{3} \times 3^{2} \times 5^{-two}}{4^{three} \times ii^{two}}\)
• \(\frac{five \times 3^{2}}{4^{three} \times iv}\)
• 45/4^four = 45/256

How to Solve Negative Exponents?

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

Example: Solve: (7³) × (3^-4/21^-2)

Solution:

Starting time, nosotros catechumen all the negative exponents to positive exponents and and so simplify

• Given: \(\frac{seven^{iii} \times 3^{-4}}{21^{-two}}\)
• Convert the negative exponents to positive by writing the reciprocal of the detail number:\(\frac{7^{3} \times 21^{2}}{3^{iv}}\)
• Utilise the rule: (ab)^{due north} = aⁿ × bⁿ and separate the required number (21).
• \(\frac{7^{3} \times seven^{2} \times 3^{ii}}{3^{4}}\)
• Use the rule: a^m × aⁿ = a^(yard+n) to combine the common base (seven).
• vii⁵/iiiⁱⁱ =16807/9

Important Notes:

Note the following points which should be remembered while we work with negative exponents.

• Exponent or power ways the number of times the base needs to exist multiplied by itself.
  a^m = a × a × a ….. thou times
  a^-m = one/a × 1/a × 1/a ….. g times
• a^-n is likewise known as the multiplicative inverse of aⁿ.
• If a^-g = a^-n then yard = north.
• The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as a^x=1/a^-ten

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. Instance 1: Find the solution of the given problem (3² + 4ⁱⁱ)^-2 past using negative exponents rules.

   Solution:

   (3^two + 4²)^-ii = (9 + xvi)^-2 = (25)^-ii = 1/25² = ane/625. Therefore, (3² + iv^two)^-2 = 1/625

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

   Solution:

   Here we take negative exponents with variables.

   27/3^-ten = 3^{half dozen}
   three³/3^-x = 3⁶
   iii^three × 3^x = 3^vi
   3^(iii+ten) = three⁶

   If bases are aforementioned then exponents must exist equal, then, ten + three = six, x = 3. Therefore, the value of x = 3.

3. Example 3: Simplify the following using negative exponent rules: (2/3)^-ii + (v)^-1

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

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Practice Questions on Negative Exponents

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FAQs on Negative Exponents

What are Negative Exponents?

When we take negative numbers as exponents, we call them negative exponents. For example, in the number 2^-8, -viii is the negative exponent of base ii.

Do negative exponents make negative numbers?

This is non true that negative exponents give negative numbers. Existence positive or negative depends on the base of the number. Negative numbers give a negative outcome when their exponent is odd and they give a positive effect when the exponent is even. For example, (-5)³ = -125, (-5)⁴ = 625. A positive number with a negative exponent will always give a positive number. For example, 2^-3 = i/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: 3^-iii + 1/2^-4, first we change these to their reciprocal course: 1/3³ + two^iv, then simplify 1/27 + 16. Taking the LCM, [1+ (xvi × 27)]/27 = 433/27.

What is the Rule for Negative Exponents?

At that place are ii main negative exponent rules that are given beneath:

• Let a be the base and n exist the exponent, we have, a^-n = 1/aⁿ.
• 1/a^-north = aⁿ

How to Divide Negative Exponents?

Dividing exponents with the same base is the aforementioned as multiplying exponents, simply commencement, 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 utilize the aforementioned rule while dividing exponents. For instance, to solve y⁵ ÷ y^-3, or, y^five/y^-three, outset we change the negative exponent (y^-iii) to a positive 1 by writing its reciprocal. This makes information technology: y⁵ × y³ = y⁽⁵⁺³⁾ = y⁸.

How to Multiply Negative Exponents?

While multiplying negative exponents, first nosotros demand to convert them to positive exponents past writing the respective numbers in their reciprocal form. Once they are converted to positive ones, nosotros multiply them using the same rules that we use for multiplying positive exponents. For instance, y^-5 × y^-2 = ane/y⁵ × i/yⁱⁱ = 1/y^(5+two) = i/y⁷.

Why are Negative Exponents Reciprocals?

When we demand to alter a negative exponent to a positive one, nosotros are supposed to write the reciprocal of the given number. So, the negative sign on an exponent indirectly means the reciprocal of the given number, in the same way equally a positive exponent means the repeated multiplication of the base.

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents tin be solved past taking the reciprocal of the fraction. Then, detect the value of the number by taking the positive value of the given negative exponent. For example, (iii/4)^-two can be solved by taking the reciprocal of the fraction, which is iv/3. At present, find the positive exponent value of 4/three, which is (four/three)² = iv²/3^two. This results in xvi/ix which is the final answer.

What is 10 to the Negative Power of ii?

ten to the negative power of two is represented every bit 10^-2, which is equal to (i/10ⁱⁱ) = i/100.

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

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