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:

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

Question 3

Solve the following equation:

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

Question 4

Simplify the following:

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

Question 5

Insert the corresponding expression:

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

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:

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 #3

Solve the following equation:

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

Video Solution

Step-by-Step Solution

We will apply the law of exponents for division between terms with identical bases:

cmcn=cm−n \frac{c^m}{c^n}=c^{m-n} Note: This law is only effective 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 #4

Simplify the following:

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

Video Solution

Step-by-Step Solution

Let's start with multiplying the fractions, remembering that multiplication of fractions is performed by multiplying numerator by numerator and denominator by denominator:

a20ba15b⋅a3ba2b=a20b⋅a3ba15b⋅a2b \frac{a^{20b}}{a^{15b}}\cdot\frac{a^{3b}}{a^{2b}}=\frac{a^{20b}\cdot a^{3b}}{a^{15b}\cdot a^{2b}}

Next, we'll note that both in the numerator and denominator, multiplication occurs between terms with identical bases, so we'll use the power law for multiplying terms with identical bases:

cm⋅cn=cm+n c^m\cdot c^n=c^{m+n}

We emphasize that this law can only be used when multiplication is performed between terms with identical bases.

From this point forward, we will no longer use the multiplication sign, but instead use the conventional notation where placing terms next to each other implies multiplication.
Let's return to the problem and apply the above power law separately to the fraction's numerator and denominator:

a20ba3ba15ba2b=a20b+3ba15b+2b=a23ba17b \frac{a^{20b}a^{3b}}{a^{15b}a^{2b}}=\frac{a^{20b+3b}}{a^{15b+2b}}=\frac{a^{23b}}{a^{17b}}

where in the final step we calculated the sum of the exponents in the numerator and denominator.

Now we notice that we need to perform division between two terms with identical bases, so we'll use the power law for dividing terms with identical bases:

cmcn=cm−n \frac{c^m}{c^n}=c^{m-n}

We emphasize that this law can only be used when division is performed between terms with identical bases.

Let's return to the problem and apply the above power law:

a23ba17b=a23b−17b=a6b \frac{a^{23b}}{a^{17b}}=a^{23b-17b}=a^{6b}

where in the final step we calculated the subtraction between the exponents.

We have obtained the most simplified expression and therefore we are done.

Therefore, the correct answer is D.

Answer

a6b a^{6b}

Exercise #5

Insert the corresponding expression:

a4−2= a^{4-2}=

Video Solution

Answer

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

Question 1

Insert the corresponding expression:

\( a^7= \)

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

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

Question 4

Insert the corresponding expression:

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

Question 5

Insert the corresponding expression:

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

Exercise #6

Insert the corresponding expression:

a7= a^7=

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:

(20×3)6= \left(20\times3\right)^6=

Video Solution

Answer

a'+b' are correct

Exercise #9

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 #10

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:

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

Question 2

Insert the corresponding expression:

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

Question 3

Insert the corresponding expression:

\( x^2= \)

Question 4

Insert the corresponding expression:

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

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:

(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 #12

Insert the corresponding expression:

x10−5= x^{10-5}=

Video Solution

Answer

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

Exercise #13

Insert the corresponding expression:

x2= x^2=

Video Solution

Answer

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

Exercise #14

Insert the corresponding expression:

y6−3= y^{6-3}=

Video Solution

Answer

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

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}

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