what is 7 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 case, threeⁱⁱ = 3 × iii. In the case of positive exponents, nosotros easily multiply the number (base) by itself, but what happens when we have negative numbers every bit exponents? A negative exponent is defined as the multiplicative changed of the base, raised to the power which is opposite to the given ability. In simple words, we write the reciprocal of the number and so solve it like positive exponents. For example, (two/three)^-2 can be written as (3/2)ⁱⁱ.

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

What are Negative Exponents?

We know that the exponent of a number tells u.s. how many times we should multiply the base of operations. For example, consider 8ⁱⁱ, 8 is the base, and 2 is the exponent. We know that eight² = 8 × 8. A negative exponent tells us, how many times we have to multiply the reciprocal of the base. Consider the 8^-2, here, the base is 8 and nosotros take a negative exponent (-2). eight^-2 is expressed as ane/8ⁱⁱ = 1/viii×one/8.

Numbers and Expressions with Negative Exponents

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

Negative Exponent	Outcome
2-i	1/2
iii-2	1/32=one/9
x-three	one/x3
(2 + 4x)-two	ane/(2+4x)2
(xtwo+ yii)-3	one/(x2+y2)3

Negative Exponent Rules

Nosotros have a set of rules or laws for negative exponents which make the procedure of simplification easy. Given below are the basic rules for solving negative exponents.

• Dominion 1: The negative exponent rule states that for every number 'a' with the negative exponent -north, accept the reciprocal of the base and multiply it according to the value of the exponent: a^{(-due north)}=ane/a^north=ane/a×1/a×....n 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 event can be written equally: 1/a^(-n)=a^north=a×a×....n times

Allow us apply these rules and see how they work with numbers.

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

Solution:

• Use the negative exponent rule a^-n=one/aⁿ
• two^-2 + 3^-2 = ane/two^two + 1/iii^two = ane/four + i/9
• Take the Least Common Multiple (LCM): (nine+4)/36 = xiii/36

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

Example two: Solve: ane/4^-2 + i/2^-3

Solution:

• Apply the second rule with a negative exponent in the denominator: i/a^-n =aⁿ
• 1/4^-2 + 1/2^-3 = 4ⁱⁱ + 2³ =xvi + 8 = 24

Therefore, i/iv^-2 + ane/2^-three = 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 five^-3 becomes 1/5^three = 1/125. This is how negative exponents modify the numbers to fractions. Allow united states of america accept some other example to see how negative exponents change to fractions.

Example: Solve ii^-1 + iv^-two

Solution:

two^-1 tin can be written as 1/2 and 4^-2 is written as 1/4^two. Therefore, negative exponents go inverse 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. Equally we have already discussed that negative exponents can be expressed as fractions, then they tin easily be solved afterward they are converted to fractions. After this conversion, we multiply negative exponents using the same rules that we apply for multiplying positive exponents. Let's understand the multiplication of negative exponents with the following example.

Example: Solve: (4/v)^-3 × (10/3)^-2

• The first step is to write the expression in its reciprocal form, which changes the negative exponent to a positive one: (5/4)3×(3/ten)2
• At present open up the brackets: \(\frac{5^{3} \times 3^{2}}{4^{3} \times ten^{ii}}\)(∵10²=(5×2)² =5ⁱⁱ×2ⁱⁱ)
• Cheque the common base of operations and simplify: \(\frac{5^{3} \times 3^{two} \times v^{-ii}}{4^{3} \times two^{2}}\)
• \(\frac{v \times 3^{2}}{4^{3} \times four}\)
• 45/4^iv = 45/256

How to Solve Negative Exponents?

Solving whatsoever 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 so applying the given arithmetic operations.

Example: Solve: (seven³) × (iii^-4/21^-2)

Solution:

First, we catechumen all the negative exponents to positive exponents and then simplify

• Given: \(\frac{vii^{3} \times 3^{-4}}{21^{-ii}}\)
• Catechumen the negative exponents to positive by writing the reciprocal of the particular number:\(\frac{vii^{3} \times 21^{ii}}{3^{four}}\)
• Use the rule: (ab)ⁿ = a^northward × b^{due north} and split up the required number (21).
• \(\frac{7^{3} \times 7^{two} \times three^{2}}{3^{4}}\)
• Use the rule: a^m × a^{due north} = a^(thousand+n) to combine the common base (7).
• seven⁵/iii^two =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 of operations needs to be multiplied past itself.
  a^m = a × a × a ….. m times
  a^-m = 1/a × i/a × 1/a ….. one thousand times
• a^-n is too known as the multiplicative inverse of a^northward.
• If a^-m = a^-n then one thousand = due north.
• The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as a^x=ane/a^-10

Topics Related to Negative Exponents

Check the given articles 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: Detect the solution of the given problem (3² + 4²)^-2 past using negative exponents rules.

   Solution:

   (iii^two + 4²)^-2 = (9 + 16)^-ii = (25)^-ii = i/25^two = 1/625. Therefore, (3² + 4²)^-two = 1/625

2. Instance ii: Discover the value of ten in 27/iii^-ten = 3⁶

   Solution:

   Hither we have negative exponents with variables.

   27/3^-x = 3⁶
   3^three/3^-x = three⁶
   3^three × 3^ten = iii^vi
   iii^(iii+10) = 3⁶

   If bases are aforementioned then exponents must be equal, then, x + iii = half-dozen, x = 3. Therefore, the value of 10 = 3.

3. Example 3: Simplify the following using negative exponent rules: (2/3)^-two + (5)^-1

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

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Practise Questions on Negative Exponents

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FAQs on Negative Exponents

What are Negative Exponents?

When we have negative numbers every bit exponents, we call them negative exponents. For example, in the number two^-8, -eight is the negative exponent of base 2.

Do negative exponents make negative numbers?

This is not truthful that negative exponents give negative numbers. Beingness 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 effect when the exponent is fifty-fifty. For instance, (-five)³ = -125, (-5)^four = 625. A positive number with a negative exponent will e'er requite a positive number. For example, 2^-3 = 1/viii, 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^-three + ane/2^-iv, first we change these to their reciprocal class: 1/3³ + 2⁴, and so simplify 1/27 + xvi. Taking the LCM, [ane+ (16 × 27)]/27 = 433/27.

What is the Dominion for Negative Exponents?

In that location are two primary negative exponent rules that are given below:

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

How to Divide Negative Exponents?

Dividing exponents with the same base is the same equally multiplying exponents, but first, we demand 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 aforementioned rule while dividing exponents. For example, to solve y⁵ ÷ y^-3, or, y⁵/y^-3, offset we change the negative exponent (y^-three) to a positive one by writing its reciprocal. This makes it: y⁵ × y³ = y⁽⁵⁺³⁾ = y^viii.

How to Multiply Negative Exponents?

While multiplying negative exponents, beginning we need to catechumen them to positive exponents by writing the corresponding numbers in their reciprocal grade. Once they are converted to positive ones, we multiply them using the same rules that we apply for multiplying positive exponents. For case, y^-5 × y^-two = 1/y^five × 1/y² = one/y⁽⁵⁺ⁱⁱ⁾ = ane/y^seven.

Why are Negative Exponents Reciprocals?

When we need to change a negative exponent to a positive i, we 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 as a positive exponent ways the repeated multiplication of the base of operations.

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents can exist solved by taking the reciprocal of the fraction. Then, discover the value of the number by taking the positive value of the given negative exponent. For example, (iii/iv)^-ii can be solved past taking the reciprocal of the fraction, which is 4/3. Now, find the positive exponent value of iv/iii, which is (4/iii)^two = 4²/3ⁱⁱ. This results in 16/nine which is the final answer.

What is x to the Negative Ability of 2?

10 to the negative ability of 2 is represented equally x^-2, which is equal to (1/10²) = one/100.

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

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