1
Solved example of quotient of powers
$\frac{\left(\left(3x y^2\right)^4\left(2x^3 y^4\right)^3\right)^2}{\left(4x^2 y^3\right)^5}$
2
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(\left(3xy^2\right)^4\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2y^3\right)^5}$
3
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(\left(3xy^2\right)^4\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
4
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(\left(3xy^2\right)^4\right)^2\left(\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
5
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\left(xy^2\right)^4\right)^2\left(\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
6
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(\left(xy^2\right)^4\right)^2\left(\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
7
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(x^4\left(y^2\right)^4\right)^2\left(\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
8
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(\left(2x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
9
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\left(x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
10
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\right)^2\left(\left(x^3y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
11
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\left(3^4\right)^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{4^5\left(x^2\right)^5\left(y^3\right)^5}$
12
Calculate the power $4^5$
$\frac{\left(3^4\right)^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
13
Calculate the power $3^4$
$\frac{81^2\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
14
Calculate the power $81^2$
$\frac{6561\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(2^3\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
15
Calculate the power $2^3$
$\frac{6561\left(x^4\right)^2\left(\left(y^2\right)^4\right)^28^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
16
Calculate the power $8^2$
$\frac{6561\cdot 64\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
17
Multiply $6561$ times $64$
$\frac{419904\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024\left(x^2\right)^5\left(y^3\right)^5}$
18
Applying the power of a power property
$\frac{\frac{6561}{16}x^{8}y^{16}\left(x^{9}y^{12}\right)^2}{x^{10}y^{15}}$
19
Simplify the fraction $\frac{\frac{6561}{16}x^{8}y^{16}\left(x^{9}y^{12}\right)^2}{x^{10}y^{15}}$ by $y$
$\frac{\frac{6561}{16}x^{8}y\left(x^{9}y^{12}\right)^2}{x^{10}}$
20
Simplify the fraction by $x$
$\frac{\frac{6561}{16}y\left(x^{9}y^{12}\right)^2}{x^{2}}$
21
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\frac{6561}{16}y\left(x^{9}\right)^2\left(y^{12}\right)^2}{x^{2}}$
22
Applying the power of a power property
$\frac{\frac{6561}{16}yx^{18}y^{24}}{x^{2}}$
23
When multiplying exponents with same base you can add the exponents: $\frac{6561}{16}yx^{18}y^{24}$
$\frac{\frac{6561}{16}y^{25}x^{18}}{x^{2}}$
24
Simplify the fraction $\frac{\frac{6561}{16}y^{25}x^{18}}{x^{2}}$ by $x$
$\frac{6561}{16}y^{25}x^{16}$
Final Answer
$\frac{6561}{16}y^{25}x^{16}$