Power of a Quotient Rule for Exponents: Inverse formula

Solving the problemCombination of different basesNumbers as coefficientsCalculating powers with negative exponentsIdentify the greater valuePresenting powers with negative exponents as fractionsNumber of termsVariables in the exponent of the powerUsing multiple rulesVariable in the base of the powerSystem of equations with no solutionApplying the formula
Question 1

Insert the corresponding expression:

\( 3^{x-y}= \)

Question 2

Insert the corresponding expression:

\( 20^{6y}= \)

Question 3

Insert the corresponding expression:

\( \left(2\times5\right)^{9-2}= \)

Question 4

Insert the corresponding expression:

\( \left(6\times8\right)^{12-6}= \)

Question 5

Insert the corresponding expression:

\( \left(4\times9\right)^5= \)

Examples with solutions for Power of a Quotient Rule for Exponents: Inverse formula

Exercise #1

Insert the corresponding expression:

3x−y= 3^{x-y}=

Video Solution

Step-by-Step Solution

To solve the problem, we need to understand the Power of a Quotient Rule for Exponents. The rule states that:

  • If you have a power of the quotient, such as: (ab)n (\frac{a}{b})^n , it can also be expressed as: anbn \frac{a^n}{b^n}

In the given expression, we have 3x−y 3^{x-y} . This can be rewritten using the inverse of the rule as:

3x−y=3x3y 3^{x-y} = \frac{3^x}{3^y}

Here’s a step-by-step breakdown:

  1. Start with the expression: 3x−y 3^{x-y} .
  2. Using the law of exponents, which states that am−n=aman a^{m-n} = \frac{a^m}{a^n} , we rewrite the expression.
  3. Replace a a with 3 3 , m m with x x , and n n with y y to get: 3x−y=3x3y 3^{x-y} = \frac{3^x}{3^y} .

The solution to the question is: 3x3y \frac{3^x}{3^y}

Answer

3x3y \frac{3^x}{3^y}

Exercise #2

Insert the corresponding expression:

206y= 20^{6y}=

Step-by-Step Solution

To solve the given expression 206y 20^{6y} and express it as a fraction aman \frac{a^m}{a^n} , we can use the Power of a Quotient Rule for Exponents. This rule states that:

  • aman=am−n \frac{a^m}{a^n} = a^{m-n}

We have the expression 206y 20^{6y} which we want to represent as 20m20n \frac{20^{m}}{20^{n}} .

To achieve this, we must have:

  • m−n=6y m - n = 6y

A straightforward way to do this is to choose m=10y m = 10y and n=4y n = 4y , such that:

  • 10y−4y=6y 10y - 4y = 6y

This choice satisfies the equation m−n=6y m - n = 6y , thus the given expression can be rewritten using the power of a quotient rule as:

206y=2010y204y 20^{6y} = \frac{20^{10y}}{20^{4y}}

This confirms that expressing 206y 20^{6y} as 2010y204y \frac{20^{10y}}{20^{4y}} is consistent with the given correct answer.

The solution to the question is: 2010y204y \frac{20^{10y}}{20^{4y}}

Answer

2010y204y \frac{20^{10y}}{20^{4y}}

Exercise #3

Insert the corresponding expression:

(2×5)9−2= \left(2\times5\right)^{9-2}=

Video Solution

Answer

(2×5)9(2×5)2 \frac{\left(2\times5\right)^9}{\left(2\times5\right)^2}

Exercise #4

Insert the corresponding expression:

(6×8)12−6= \left(6\times8\right)^{12-6}=

Video Solution

Answer

(6×8)12(6×8)6 \frac{\left(6\times8\right)^{12}}{\left(6\times8\right)^6}

Exercise #5

Insert the corresponding expression:

(4×9)5= \left(4\times9\right)^5=

Video Solution

Answer

(4×9)10(4×9)5 \frac{\left(4\times9\right)^{10}}{\left(4\times9\right)^5}

Question 1

Insert the corresponding expression:

\( \left(20\times3\right)^6= \)

Question 2

Insert the corresponding expression:

\( \left(15\times5\right)^3= \)

Question 3

Insert the corresponding expression:

\( a^{4-2}= \)

Question 4

Insert the corresponding expression:

\( x^{10-5}= \)

Question 5

Insert the corresponding expression:

\( y^{6-3}= \)

Exercise #6

Insert the corresponding expression:

(20×3)6= \left(20\times3\right)^6=

Video Solution

Answer

a'+b' are correct

Exercise #7

Insert the corresponding expression:

(15×5)3= \left(15\times5\right)^3=

Video Solution

Answer

(15×5)8(15×5)5 \frac{\left(15\times5\right)^8}{\left(15\times5\right)^5}

Exercise #8

Insert the corresponding expression:

a4−2= a^{4-2}=

Video Solution

Answer

a4a2 \frac{a^4}{a^2}

Exercise #9

Insert the corresponding expression:

x10−5= x^{10-5}=

Video Solution

Answer

x10x5 \frac{x^{10}}{x^5}

Exercise #10

Insert the corresponding expression:

y6−3= y^{6-3}=

Video Solution

Answer

y6y3 \frac{y^6}{y^3}

Question 1

Insert the corresponding expression:

\( x^2= \)

Question 2

Insert the corresponding expression:

\( a^7= \)

Question 3

Insert the corresponding expression:

\( \left(7\times2\right)^{2y-a-2}= \)

Question 4

Insert the corresponding expression:

\( \left(10\times5\right)^{4y-x}= \)

Question 5

Insert the corresponding expression:

\( \left(6\times9\right)^{10y}= \)

Exercise #11

Insert the corresponding expression:

x2= x^2=

Video Solution

Answer

x4x2 \frac{x^4}{x^2}

Exercise #12

Insert the corresponding expression:

a7= a^7=

Video Solution

Answer

A+B are correct

Exercise #13

Insert the corresponding expression:

(7×2)2y−a−2= \left(7\times2\right)^{2y-a-2}=

Video Solution

Answer

a'+b' are correct

Exercise #14

Insert the corresponding expression:

(10×5)4y−x= \left(10\times5\right)^{4y-x}=

Video Solution

Answer

(10×5)4y(10×5)x \frac{\left(10\times5\right)^{4y}}{\left(10\times5\right)^x}

Exercise #15

Insert the corresponding expression:

(6×9)10y= \left(6\times9\right)^{10y}=

Video Solution

Answer

A'+C' are correct

Question 1

Insert the corresponding expression:

\( \left(10\times4\right)^{5a}= \)

Question 2

Insert the corresponding expression:

\( 8^{ax-4}= \)

Question 3

Insert the corresponding expression:

\( 10^{2x-a-1}= \)

Question 4

Insert the corresponding expression:

\( 4^{3a}= \)

Question 5

Insert the corresponding expression:

\( \left(4\times3\right)^{4x-y}= \)

Exercise #16

Insert the corresponding expression:

(10×4)5a= \left(10\times4\right)^{5a}=

Video Solution

Answer

(10×4)8a(10×4)3a \frac{\left(10\times4\right)^{8a}}{\left(10\times4\right)^{3a}}

Exercise #17

Insert the corresponding expression:

8ax−4= 8^{ax-4}=

Video Solution

Answer

8ax84 \frac{8^{ax}}{8^4}

Exercise #18

Insert the corresponding expression:

102x−a−1= 10^{2x-a-1}=

Video Solution

Answer

102x10a+1 \frac{10^{2x}}{10^{a+1}}

Exercise #19

Insert the corresponding expression:

43a= 4^{3a}=

Video Solution

Answer

47a44a \frac{4^{7a}}{4^{4a}}

Exercise #20

Insert the corresponding expression:

(4×3)4x−y= \left(4\times3\right)^{4x-y}=

Video Solution

Answer

(4×3)4x(4×3)y \frac{\left(4\times3\right)^{4x}}{\left(4\times3\right)^y}