Power of a Quotient Rule for Exponents: Variable in the base of the power

Solving the problemCombination of different basesNumbers as coefficientsNumber of termsSystem of equations with no solutionIdentify the greater valueVariables in the exponent of the powerInverse formulaApplying the formulaUsing multiple rules
Question 1

Simplify the following:

\( \frac{a^a}{a^b}= \)

Question 2

Solve the following:

\( \frac{a^x}{a^y}+\frac{a^2}{a^x}= \)

Question 3

Solve the following:

\( \frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}= \)

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Examples with solutions for Power of a Quotient Rule for Exponents: Variable in the base of the power

Exercise #1

Simplify the following:

aaab= \frac{a^a}{a^b}=

Video Solution

Step-by-Step Solution

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

cmcn=cm−n \frac{c^m}{c^n}=c^{m-n} Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the mentioned power property:

aaab=aa−b \frac{a^a}{a^b}=a^{a-b} Therefore, the correct answer is option D.

Answer

aa−b a^{a-b}

Exercise #2

Solve the following:

axay+a2ax= \frac{a^x}{a^y}+\frac{a^2}{a^x}=

Video Solution

Step-by-Step Solution

Note that we need to perform division between two terms with identical bases, therefore we will use the law of exponents for division between terms with identical bases:

cmcn=cm−n \frac{c^m}{c^n}=c^{m-n} We emphasize that using this law is only possible when the division is between terms with identical bases.

Let's return to the problem and apply the above law of exponents to each term in the sum separately:

axay+a2ax=ax−y+a2−x \frac{a^x}{a^y}+\frac{a^2}{a^x}=a^{x-y}+a^{2-x} Therefore the correct answer is A.

Answer

ax−y+a2−x a^{x-y}+a^{2-x}

Exercise #3

Solve the following:

bybx−bzb3= \frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=

Video Solution

Step-by-Step Solution

Here we have division between two terms with identical bases, therefore we will use the power property to divide terms with identical bases:

cmcn=cm−n \frac{c^m}{c^n}=c^{m-n} Note that using this property is only possible when the division is carried out between terms with identical bases.

Let's go back to the problem and apply the power property to each term of the exercise separately:

bybx−bzb3=by−x−bz−3 \frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=b^{y-x}-b^{z-3} Therefore, the correct answer is option A.

Answer

by−x−bz−3 b^{y-x}-b^{z-3}

Exercise #4

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 #5

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}

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #6

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}

Exercise #7

Insert the corresponding expression:

(20×3)6= \left(20\times3\right)^6=

Video Solution

Answer

a'+b' are correct

Exercise #8

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 #9

Insert the corresponding expression:

a4−2= a^{4-2}=

Video Solution

Answer

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

Exercise #10

Insert the corresponding expression:

x10−5= x^{10-5}=

Video Solution

Answer

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

Question 1

Insert the corresponding expression:

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

Question 2

Insert the corresponding expression:

\( x^2= \)

Question 3

Insert the corresponding expression:

\( a^7= \)

Question 4

Simplify the following:

\( \frac{a^{20b}}{a^{15b}}\times\frac{a^{3b}}{a^{2b}}= \)

Question 5

Insert the corresponding expression:

\( \frac{\left(2\times a\right)^2}{\left(2\times a\right)^4}= \)

Exercise #11

Insert the corresponding expression:

y6−3= y^{6-3}=

Video Solution

Answer

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

Exercise #12

Insert the corresponding expression:

x2= x^2=

Video Solution

Answer

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

Exercise #13

Insert the corresponding expression:

a7= a^7=

Video Solution

Answer

A+B are correct

Exercise #14

Simplify the following:

a20ba15b×a3ba2b= \frac{a^{20b}}{a^{15b}}\times\frac{a^{3b}}{a^{2b}}=

Video Solution

Answer

a6b a^{6b}

Exercise #15

Insert the corresponding expression:

(2×a)2(2×a)4= \frac{\left(2\times a\right)^2}{\left(2\times a\right)^4}=

Video Solution

Answer

1(2×a)2 \frac{1}{\left(2\times a\right)^2}

Question 1

Insert the corresponding expression:

\( \frac{\left(a\times b\right)^x}{\left(a\times b\right)^{3x}}= \)

Question 2

Insert the corresponding expression:

\( \frac{\left(a\times x\right)^y}{\left(a\times x\right)^b}= \)

Question 3

Insert the corresponding expression:

\( \frac{\left(xya\right)^{2x}}{\left(xya\right)^{ab}}= \)

Exercise #16

Insert the corresponding expression:

(a×b)x(a×b)3x= \frac{\left(a\times b\right)^x}{\left(a\times b\right)^{3x}}=

Video Solution

Answer

1(a×b)2x \frac{1}{\left(a\times b\right)^{2x}}

Exercise #17

Insert the corresponding expression:

(a×x)y(a×x)b= \frac{\left(a\times x\right)^y}{\left(a\times x\right)^b}=

Video Solution

Answer

(a×x)y−b \left(a\times x\right)^{y-b}

Exercise #18

Insert the corresponding expression:

(xya)2x(xya)ab= \frac{\left(xya\right)^{2x}}{\left(xya\right)^{ab}}=

Video Solution

Answer

(xya)2x−ab \left(xya\right)^{2x-ab}