What Is 10 to the Power of Negative 2

Negative Exponents

Negative exponents tell us that the ability 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, 3² = iii × 3. In the case of positive exponents, nosotros hands multiply the number (base) past itself, but what happens when we have negative numbers as exponents? A negative exponent is defined every bit the multiplicative inverse of the base of operations, raised to the power which is opposite to the given power. In unproblematic words, we write the reciprocal of the number and then solve information technology like positive exponents. For example, (2/3)^-2 can be written as (3/2)².

1.	What are Negative Exponents?
2.	Negative Exponent Rules
iii.	Why are Negative Exponents Fractions?
four.	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 us how many times we should multiply the base. For example, consider 8², 8 is the base, and 2 is the exponent. Nosotros know that 8^two = 8 × viii. A negative exponent tells us, how many times we accept to multiply the reciprocal of the base. Consider the eight^-ii, hither, the base of operations is 8 and we have a negative exponent (-ii). 8^-two is expressed as i/8² = 1/8×1/8.

Numbers and Expressions with Negative Exponents

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

Negative Exponent	Result
2-1	ane/2
3-2	ane/32=1/nine
x-3	1/ten3
(2 + 4x)-2	1/(2+4x)ii
(tenii+ y2)-3	1/(x2+yii)three

Negative Exponent Rules

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

• Rule ane: 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 according to the value of the exponent: a^(-n)=ane/aⁿ=1/a×1/a×....northward 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 exist written as: 1/a^(-northward)=aⁿ=a×a×....n times

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

Case 1: Solve: 2^-2 + 3^-two

Solution:

• Use the negative exponent rule a^-n=1/a^{due north}
• 2^-2 + 3^-ii = 1/2² + ane/3² = 1/4 + 1/9
• Take the Least Common Multiple (LCM): (9+four)/36 = 13/36

Therefore, two^-2 + 3^-ii = xiii/36

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

Solution:

• Apply the second rule with a negative exponent in the denominator: ane/a^-n =a^northward
• 1/4^-2 + i/2^-three = iv² + 2^three =xvi + 8 = 24

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

Why are Negative Exponents Fractions?

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

Example: Solve two^-1 + 4^-two

Solution:

ii^-1 can be written as 1/2 and 4^-2 is written every bit 1/four². Therefore, negative exponents get changed to fractions when the sign of their exponent changes.

Multiplying Negative Exponents

Multiplication of negative exponents is the same as the multiplication of whatsoever other number. As nosotros have already discussed that negative exponents can be expressed as fractions, then they can hands be solved afterward they are converted to fractions. After this conversion, we multiply negative exponents using the aforementioned rules that we utilize for multiplying positive exponents. Let's understand the multiplication of negative exponents with the following example.

Case: Solve: (4/five)^-3 × (10/iii)^-2

• The starting time step is to write the expression in its reciprocal grade, which changes the negative exponent to a positive one: (v/4)3×(3/10)2
• Now open the brackets: \(\frac{5^{3} \times 3^{two}}{iv^{3} \times ten^{2}}\)(∵10ⁱⁱ=(5×2)² =5²×2²)
• Check the common base and simplify: \(\frac{5^{3} \times 3^{2} \times 5^{-two}}{iv^{iii} \times 2^{2}}\)
• \(\frac{five \times 3^{2}}{iv^{iii} \times 4}\)
• 45/4⁴ = 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 nigh the simplification of terms with negative exponents so applying the given arithmetic operations.

Example: Solve: (7³) × (3^-4/21^-2)

Solution:

Beginning, nosotros convert all the negative exponents to positive exponents and then simplify

• Given: \(\frac{seven^{3} \times 3^{-four}}{21^{-ii}}\)
• Convert the negative exponents to positive by writing the reciprocal of the particular number:\(\frac{7^{3} \times 21^{ii}}{3^{iv}}\)
• Apply the rule: (ab)ⁿ = aⁿ × bⁿ and split the required number (21).
• \(\frac{7^{3} \times vii^{ii} \times 3^{2}}{3^{iv}}\)
• Use the rule: a^m × aⁿ = a^(m+n) to combine the common base (7).
• seven⁵/three² =16807/9

Important Notes:

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

• Exponent or ability ways the number of times the base needs to exist multiplied by itself.
  a^k = a × a × a ….. thou times
  a^-m = 1/a × 1/a × i/a ….. m times
• a^-n is besides known equally the multiplicative inverse of aⁿ.
• If a^-grand = a^{-due north} then one thousand = n.
• The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as a^x=i/a^-x

Topics Related to Negative Exponents

Check the given manufactures like 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 trouble (iii² + 4²)^-2 by using negative exponents rules.

   Solution:

   (3² + 4²)^-2 = (9 + 16)^-2 = (25)^-2 = one/25² = 1/625. Therefore, (3² + fourⁱⁱ)^-2 = one/625

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

   Solution:

   Hither we have negative exponents with variables.

   27/3^-ten = three⁶
   3³/3^-ten = 3^half-dozen
   3³ × three^x = 3⁶
   3^(iii+ten) = iii⁶

   If bases are same then exponents must be equal, so, x + 3 = 6, ten = 3. Therefore, the value of x = 3.

3. Example 3: Simplify the following using negative exponent rules: (2/3)^-2 + (five)^-ane

   Solution: By using negative exponent rules, we tin write (2/3)^-2 as (iii/2)² and (five)^-1 as i/5. And then, we tin can simplify the given expression as,
   (three/two)^two + 1/5
   nine/iv + one/5
   (45+4)/twenty
   49/twenty
   Therefore, (two/3)^-two + (v)^-ane 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 accept negative numbers as exponents, nosotros call them negative exponents. For instance, in the number ii^-8, -8 is the negative exponent of base two.

Do negative exponents brand negative numbers?

This is non truthful that negative exponents give negative numbers. Existence positive or negative depends on the base of the number. Negative numbers requite a negative issue when their exponent is odd and they give a positive outcome when the exponent is even. For instance, (-five)³ = -125, (-5)^iv = 625. A positive number with a negative exponent will always requite a positive number. For example, 2^-three = i/8, which is a positive number.

How to Simplify Negative Exponents?

Negative exponents are simplified using the aforementioned laws of exponents that are used to solve positive exponents. For case, to solve: 3^-iii + 1/2^-4, start we change these to their reciprocal form: i/3³ + 2^iv, then simplify one/27 + 16. Taking the LCM, [1+ (16 × 27)]/27 = 433/27.

What is the Rule for Negative Exponents?

There are two main negative exponent rules that are given below:

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

How to Separate Negative Exponents?

Dividing exponents with the same base is the same as multiplying exponents, but starting time, 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 utilise the same rule while dividing exponents. For example, to solve y^five ÷ y^-3, or, y^five/y^-three, first we change the negative exponent (y^-3) to a positive one by writing its reciprocal. This makes information technology: y⁵ × yⁱⁱⁱ = y^(five+3) = y⁸.

How to Multiply Negative Exponents?

While multiplying negative exponents, first nosotros need 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 utilize for multiplying positive exponents. For example, y^-5 × y^-ii = ane/y⁵ × 1/yⁱⁱ = 1/y⁽⁵⁺²⁾ = ane/y^seven.

Why are Negative Exponents Reciprocals?

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

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents can 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, (3/four)^-ii can be solved by taking the reciprocal of the fraction, which is 4/3. Now, observe the positive exponent value of iv/3, which is (4/3)² = 4^two/iii^two. This results in 16/9 which is the final answer.

What is 10 to the Negative Power of 2?

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

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

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