Question

Simplify. Show all work to receive full credit!


4(3x²y⁴)³/(2x³y⁵)⁴
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Answer 1

`(4(3x^2y^4)^3)/((2x^3y^5)^4)=(27)/(4x^6y^8)`

Explanation:

Given:

The expression to simplify is given as:

`(4(3x^2y^4)^3)/((2x^3y^5)^4)`

First, we will simplify the numerator and denominator separately using the law of indices:

`(ka^mb^n)^p=k^pa^(m* p)b^(n* p)`

The numerator is simplified as:

`4(3x^2y^4)^3=4(3^3x^(2* 3)y^(4* 3))=4(27x^6y^(12))`

The denominator is simplified as:

`(2x^3y^5)^4=2^4x^(3* 4)y^(5* 4)=16x^(12)y^(20)`

Now, we divide the simplified numerator by the simplified denominator. This gives,

`=(4(27x^6y^(12)))/(16x^(12)y^(20))\\=(4*27)/(16)* (x^6)/(x^(12))* (y^(12))/(y^(20))`

Now, we simplify using another law of indices which is given as:

`(a^m)/(a^n)=a^(m-n)`

`=(4*27)/(16)* (x^6)/(x^(12))* (y^(12))/(y^(20))\\=(27)/(4)* x^(6-12)* y^(12-20)\\=(27)/(4)x^(-6)y^(-8)`

Now, we write the answer using only positive exponents and thus we use the given law of indices:

`a^(-m)=(1)/(a^m)`

Therefore, the simplified form is:

`(27)/(4)x^(-6)y^(-8)=(27)/(4x^6y^8)`