Negative Exponent Times Positive Exponent

Negative Exponents

A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is of the opposite sign of the given ability. In unproblematic words, we write the reciprocal of the number and then solve it similar positive exponents. For example, (2/three)^-2 can be written as (3/2)^two. Nosotros know that an exponent refers to the number of times a number is multiplied by itself. For instance, three² = 3 × 3. In the instance of positive exponents, nosotros hands multiply the number (base) by itself, just in case of negative exponents, we multiply the reciprocal of the number by itself. For case, 3^-2 = 1/3 × one/3.

Allow us learn more than virtually negative exponents along with related rules and solve more examples.

1.	What are Negative Exponents?
2.	Numbers and Expressions with Negative Exponents
iii.	Negative Exponent Rules
4.	Why are Negative Exponents Fractions?
5.	Negative Fraction Exponents
half-dozen.	Multiplying Negative Exponents
seven.	How to Solve Negative Exponents?
eight.	FAQs on Negative Exponents

What are Negative Exponents?

We know that the exponent of a number tells us how many times nosotros should multiply the base. For example, in 8², 8 is the base, and ii is the exponent. We know that 8² = eight × 8. A negative exponent tells u.s., how many times we have to multiply the reciprocal of the base. Consider the 8^-2, hither, the base of operations is viii and we have a negative exponent (-two). eight^-two is expressed equally ane/viii × one/eight = 1/viii².

Numbers and Expressions with Negative Exponents

Here are a few examples which limited negative exponents with variables and numbers. Observe the tabular array 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-ane	1/2
three-ii	1/three2 = 1/9
x-iii	one/x3
(2 + 4x)-two	i/(2 + 4x)2
(x2 + y2)-3	one/(x2 + y2)3

Negative Exponent Rules

We have a prepare of rules or laws for negative exponents which brand the procedure of simplification easy. Given below are the basic rules for solving negative exponents.

• Rule 1: The negative exponent dominion 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 n times.
  i.e., a^(-n) = 1/a × 1/a × ... northward times = 1/a^northward
• Rule 2: The rule is the same even when there is a negative exponent in the denominator.
  i.e., ane/a^(-n) = a × a × ... .n times = a^north

Permit us apply these rules and run into how they work with numbers.

Case i: Solve: 2^-2 + 3^-2

Solution:

• Use the negative exponent rule a^{-due north} = 1/a^north
• two^-ii + 3^-two = ane/ii² + 1/3² = 1/4 + one/9
• Have the Least Mutual Multiple (LCM): (ix + 4)/36 = 13/36

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

Example 2: Solve: ane/iv^-2 + 1/two^-3

Solution:

• Utilise the second rule with a negative exponent in the denominator: ane/a^-north =aⁿ
• 1/4^-2 + i/two^-3 = 4² + two³ =16 + 8 = 24

Therefore, 1/iv^-two + 1/2^-3 = 24.

Negative Exponents are Fractions

A negative exponent takes us to the inverse of the number. In other words, a^-northward = 1/aⁿ and v^-three becomes 1/5³ = ane/125. This is how negative exponents change the numbers to fractions. Allow us take another instance to come across how negative exponents change to fractions.

Example: Express 2^-1 and 4^-2 as fractions.

Solution:

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

Negative Fraction Exponents

Sometimes, we might have a negative fractional exponent like iv^-iii/2. Nosotros can apply the same rule a^-north = 1/aⁿ to express this in terms of a positive exponent. i.e., 4^-3/two = one/iv^3/2. Further, nosotros tin can simplify this using the exponent rules.

4^-iii/ii = ane/4^3/ii

= ane / (ii²)^three/2

= ane / 2³

= 1/viii

Multiplying Negative Exponents

Multiplication of negative exponents is the same equally the multiplication of any other number. As we have already discussed that negative exponents can be expressed as fractions, so they can hands be solved later on they are converted to fractions. Later this conversion, nosotros multiply negative exponents using the same multiplication dominion that we apply for multiplying positive exponents. Let u.s. empathize the multiplication of negative exponents with the following example.

Example: Solve: (four/five)^-3 × (10/3)^-2

• The offset step is to write the expression in its reciprocal form, which changes the negative exponent to a positive one: (5/iv)³ × (3/10)²
• Now open up the brackets: \(\frac{v^{3} \times iii^{ii}}{4^{3} \times 10^{2}}\)
• We know that 10^two=(v×ii)² =5²×two², so we can substitute x² by five²×2². So we will check the common base and simplify: \(\frac{five^{three} \times three^{two} \times 5^{-2}}{four^{3} \times two^{2}}\)
• \(\frac{5 \times 3^{2}}{4^{three} \times four}\)
• 45/iv^iv = 45/256

How to Solve Negative Exponents?

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

• a^-n = 1/a^{due north}
• 1/a^-n = aⁿ

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

Solution:

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

• Given: \(\frac{7^{3} \times 3^{-four}}{21^{-2}}\)
• Catechumen the negative exponents to positive past applying the in a higher place rules:\(\frac{7^{3} \times 21^{2}}{three^{4}}\)
• Employ the dominion: (ab)ⁿ = a^{due north} × bⁿ and carve up the required number (21).
• \(\frac{7^{three} \times 7^{ii} \times 3^{2}}{three^{iv}}\)
• Use the rule: a^yard × aⁿ = a^(thou+n) to combine the common base (7).
• 7⁵/3² =16807/9

Important Notes on Negative Exponents:

• Exponent or ability means the number of times the base needs to be multiplied by itself.
  a^g = a × a × a ….. m times
  a^-m = 1/a × 1/a × i/a ….. g times
• a^-n is also known every bit the multiplicative changed of aⁿ.
• If a^-m = a^-north then chiliad = n.
• The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as a^ten=1/a^-x

☛ Related Topics:

• Negative Exponents Calculator
• Exponent Rules Calculator
• Exponent Calculator

Examples of Negative Exponents

1. Case ane: Find the solution of the given expression (iii² + fourⁱⁱ)^-2 .

   Solution:

   The given expression is,

   (3ⁱⁱ + 4^two)^-2 = (nine + 16)^-2
   = (25)^-ii
   = 1/25² (past negative exponents dominion)
   = 1/625.
   Therefore, (3² + iv^two)^-2 = 1/625

   Answer: i/625

2. Example 2: Find the value of x in 27/3^-x = 3⁶

   Solution:

   Here we have negative exponents with variables.

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

   If bases are the same and so exponents must be equal, and then, 3 + 10 = 6. Solving this, x = iii.

   Answer: x = iii.

3. Example three: Simplify the following using negative exponent rules: (ii/three)^-2 + (5)^-one

   Solution:

   Past using negative exponent rules, nosotros can write (2/3)^-2 as (3/two)^two and (five)^-1 as 1/5. So, we tin can simplify the given expression equally,
   = (3/2)² + one/5
   = 9/iv + ane/5
   After taking the LCM, nosotros become, (45 + 4)/20
   49/xx
   Therefore, (ii/3)^-two + (5)^-ane is simplified to 49/20.

   Respond: 49/20.

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Practice Questions on Negative Exponents

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FAQs on Negative Exponents

What do Negative Exponents Mean?

The negative exponents mean the negative numbers that are present in place of exponents. For example, in the number 2^-viii, -viii is the negative exponent of base ii.

Practise negative exponents Result in Negative Numbers?

No, information technology is not necessary that negative exponents give negative numbers. For case, 2^-3 = 1/8, which is a positive number.

How to Calculate Negative Exponents?

Negative exponents are calculated using the same laws of exponents that are used to solve positive exponents. For example, to solve: 3^-three + i/2^-4, first nosotros alter these to their reciprocal course: ane/iii³ + 2⁴, then simplify 1/27 + 16. Taking the LCM, [ane+ (sixteen × 27)]/27 = 433/27.

What is the Rule for Negative Exponents?

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

• a^-north = 1/aⁿ
• ane/a^-n = a^north

How to Solve Fractions with Negative Exponents?

Fractions with negative exponents can exist solved by taking the reciprocal of the fraction. And so, find the value of the number by taking the positive value of the given negative exponent. For example, (3/iv)^-2 = (iv/3)^two = 4^two/three². This results in 16/ix which is the concluding respond.

How to Divide Negative Exponents?

Dividing exponents with the same base results in the subtraction of exponents. For example, to solve y^five ÷ y^-three = y^5-(-3) = y⁸. This tin can be simplified in an alternative mode as well. i.e., y^v ÷ y^-three = y^five/y^-3, first we alter the negative exponent (y^-3) to a positive one by writing its reciprocal. This makes information technology: y⁵ × y³ = y⁽⁵⁺³⁾ = y^viii.

How to Multiply Negative Exponents?

While multiplying negative exponents, first nosotros need to convert them to positive exponents past writing the corresponding numbers in their reciprocal form. Once they are converted to positive ones, we multiply them using the aforementioned rules that we utilise for multiplying positive exponents. For instance, y^-five × y^-2 = 1/y^five × ane/y² = one/y^(five+2) = one/y^seven

Why are Negative Exponents Reciprocals?

When we need to change a negative exponent to a positive one, 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 mode as a positive exponent means the repeated multiplication of the base.

What is ten to the Negative Power of 2?

10 to the negative power of ii is represented every bit 10^-2, which is equal to (one/10²) = 1/100.

Negative Exponent Times Positive Exponent,

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

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