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

Simplify the following:

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

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

$\frac{c^m}{c^n}=c^{m-n}$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:

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

$a^{a-b}$

Solve the following:

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

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

$\frac{c^m}{c^n}=c^{m-n}$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:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=b^{y-x}-b^{z-3}$Therefore, the correct answer is option A.

$b^{y-x}-b^{z-3}$

Solve the exercise:

$a^2:a+a^3\cdot a^5=$

First we rewrite the first expression on the left of the problem as a fraction:

$\frac{a^2}{a}+a^3\cdot a^5$Then we use two properties of exponentiation, to multiply and divide terms with identical bases:

A.

$b^m\cdot b^n=b^{m+n}$2.

$\frac{b^m}{b^n}=b^{m-n}$Returning to the problem and applying the two properties of exponentiation mentioned earlier:

$\frac{a^2}{a}+a^3\cdot a^5=a^{2-1}+a^{3+5}=a^1+a^8=a+a^8$

Later on, keep in mind that we need to factor the expression we obtained in the last step by extracting the common factor,

Therefore, we extract from outside the parentheses the greatest common divisor to the two terms which are:

$a$ We obtain the expression:

$a+a^8=a(1+a^7)$when we use the property of exponentiation mentioned earlier in A.

$$$a^8=a^{1+7}=a^1\cdot a^7=a\cdot a^7$

Summarizing the solution to the problem and all the steps, we obtained the following:

$\frac{a^2}{a}+a^3\cdot a^5=a(1+a^{7)}$Therefore, the correct answer is option b.

$a(1+a^7)$

Solve the following:

$\frac{a^x}{a^y}+\frac{a^2}{a^x}=$

$a^{x-y}+a^{2-x}$

Solve the exercise:

$X^3\cdot X^2:X^5+X^4$

$1+X^4$

Simplify the following:

$\lbrack\frac{a^4}{a^3}\times\frac{a^8}{a^7}\rbrack:\frac{a^{10}}{a^8}$

$1$

Simplify the following:

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

$a^{6b}$

Simplify the following:

$\frac{b^{22}}{b^{20}}\times\frac{b^{30}}{b^{20}}=$

$b^{12}$