1
Solved example of exponent properties
$\frac{\left(\left(3x y^2\right)^4\left(2x^3 y^4\right)^3\right)^2}{\left(4x^2 y^3\right)^5}$
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
Simplify $\left(x^2\right)^5$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $5$
$\frac{419904\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}\left(y^3\right)^5}$
19
Simplify $\left(y^3\right)^5$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $5$
$\frac{419904\left(x^4\right)^2\left(\left(y^2\right)^4\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
20
Simplify $\left(x^4\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $4$ and $n$ equals $2$
$\frac{419904x^{4\cdot 2}\left(\left(y^2\right)^4\right)^2\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
21
Simplify $\left(\left(y^2\right)^4\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $4$ and $n$ equals $2$
$\frac{419904x^{4\cdot 2}\left(y^2\right)^{4\cdot 2}\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
22
Simplify $\left(y^2\right)^{4\cdot 2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $4\cdot 2$
$\frac{419904x^{4\cdot 2}y^{2\cdot 4\cdot 2}\left(\left(x^3\right)^3\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
23
Simplify $\left(x^3\right)^3$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $3$
$\frac{419904x^{4\cdot 2}y^{2\cdot 4\cdot 2}\left(x^{3\cdot 3}\left(y^4\right)^3\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
24
Simplify $\left(y^4\right)^3$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $4$ and $n$ equals $3$
$\frac{419904x^{4\cdot 2}y^{2\cdot 4\cdot 2}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{2\cdot 5}y^{3\cdot 5}}$
25
Multiply $2$ times $5$
$\frac{419904x^{4\cdot 2}y^{2\cdot 4\cdot 2}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{10}y^{3\cdot 5}}$
26
Multiply $3$ times $5$
$\frac{419904x^{4\cdot 2}y^{2\cdot 4\cdot 2}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{10}y^{15}}$
27
Multiply $4$ times $2$
$\frac{419904x^{8}y^{2\cdot 4\cdot 2}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{10}y^{15}}$
28
Multiply $2$ times $4$
$\frac{419904x^{8}y^{8\cdot 2}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{10}y^{15}}$
29
Multiply $8$ times $2$
$\frac{419904x^{8}y^{16}\left(x^{3\cdot 3}y^{4\cdot 3}\right)^2}{1024x^{10}y^{15}}$
30
Multiply $3$ times $3$
$\frac{419904x^{8}y^{16}\left(x^{9}y^{4\cdot 3}\right)^2}{1024x^{10}y^{15}}$
31
Multiply $4$ times $3$
$\frac{419904x^{8}y^{16}\left(x^{9}y^{12}\right)^2}{1024x^{10}y^{15}}$
32
Simplify the fraction $\frac{419904x^{8}y^{16}\left(x^{9}y^{12}\right)^2}{1024x^{10}y^{15}}$ by $y$
$\frac{419904x^{8}y^{\left(16-15\right)}\left(x^{9}y^{12}\right)^2}{1024x^{10}}$
33
Subtract the values $16$ and $-15$
$\frac{419904x^{8}y^{1}\left(x^{9}y^{12}\right)^2}{1024x^{10}}$
34
Any expression to the power of $1$ is equal to that same expression
$\frac{419904x^{8}y\left(x^{9}y^{12}\right)^2}{1024x^{10}}$
35
Simplify the fraction by $x$
$\frac{419904y\left(x^{9}y^{12}\right)^2}{1024x^{\left(10-8\right)}}$
36
Subtract the values $10$ and $-8$
$\frac{419904y\left(x^{9}y^{12}\right)^2}{1024x^{2}}$
37
The power of a product is equal to the product of it's factors raised to the same power
$\frac{419904y\left(x^{9}\right)^2\left(y^{12}\right)^2}{1024x^{2}}$
38
Simplify $\left(x^{9}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $9$ and $n$ equals $2$
$\frac{419904yx^{9\cdot 2}\left(y^{12}\right)^2}{1024x^{2}}$
39
Simplify $\left(y^{12}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $12$ and $n$ equals $2$
$\frac{419904yx^{9\cdot 2}y^{12\cdot 2}}{1024x^{2}}$
40
Multiply $9$ times $2$
$\frac{419904yx^{18}y^{12\cdot 2}}{1024x^{2}}$
41
Multiply $12$ times $2$
$\frac{419904yx^{18}y^{24}}{1024x^{2}}$
42
When multiplying exponents with same base you can add the exponents: $419904yx^{18}y^{24}$
$\frac{419904y^{\left(24+1\right)}x^{18}}{1024x^{2}}$
43
Add the values $24$ and $1$
$\frac{419904y^{25}x^{18}}{1024x^{2}}$
44
Simplify the fraction $\frac{419904y^{25}x^{18}}{1024x^{2}}$ by $x$
$\frac{419904y^{25}x^{\left(18-2\right)}}{1024}$
45
Subtract the values $18$ and $-2$
$\frac{419904y^{25}x^{16}}{1024}$
Final Answer
$\frac{419904y^{25}x^{16}}{1024}$