Question

Simplify the following
(-xyz^3)^6(x^2y^3z^4)^2(-3x^3yz^2)^4

Answer 1

`\boxed{\sf 81x^(22)y^(16)z^(34)}`

Explanation:

`\textsf{Our objective is to simplify the expression given to us.}`

`\large\underline{\textsf{The Idea For Our Problem.}}`

`\textsf{When we are simplifying an expression, we want to identify ways to make the}`
`\textsf{expression less complex. There are many ways to simplify expressions, especially for}`

`\textsf{the problem we're given.}`

`\large\underline{\textsf{What we should know already.}}`
`\textsf{Exponent Rules. We should know exponent rules to calculate the exponent }`

`\textsf{accurately.}`

`\sf (5^x+5^y)^z = 5^(xy)+5^(yz)`

`5^6 * 5^2} = 5^(6+2)`

`\large\underline{\textsf{Solving.}}`

`\textsf{Using our knowledge of exponent rules, we can begin simplifying the expression.}`

`\sf \boxed{\sf (-xyz^3)^6}(x^2y^3z^4)^2(-3x^3yz^2)^4`

`\textsf{Let's focus on the first set of parentheses. We know that when an exponent is}`

`\textsf{outside of the parentheses, that number represents how many times the term}`

`\textsf{will multiply to itself.}`

`\sf (-xyz^3)^6 = (-x * -x * -x * -x * -x * -x) * (y * y * y * y* y * y) * (z^3 * z^3 * z^3 * z^3 * z^3 * z^3) \rightarrow \boxed{\sf x^6y^6z^(18)}`

`\sf (-xyz^3)^6\boxed{\sf (x^2y^3z^4)^2}(-3x^3yz^2)^4`

`\textsf{Let's focus on the second set of parentheses.}`

`\sf (x^2y^3z^4)^2 = (x^2 * x^2) * (y^3 * y^3) * (z^4 * z^4) \rightarrow \boxed{\sf x^4y^6z^8}`

`\sf (-xyz^3)^6(x^2y^3z^4)^2 \boxed{\sf(-3x^3yz^2)^4}`

`\textsf{Let's focus on the last set of parentheses.}`

`\sf (-3x^3yz^2)^4 = (-3x^3 * -3x^3 * -3x^3 * -3x^3) * (y * y * y * y) * (z^2 * z^2 * z^2 * z^2) \rightarrow \boxed{\sf 81x^(12)y^4z^8}`

`\textsf{As you may already tell, we now have to multiply these expressions.}`

`\sf x^6y^6z^(18) * x^4y^6z^8 * 81x^(12)y^4z^8`

`\textsf{Continuing with exponent rules, like terms multiply together and exponents}`

`\textsf{combine.}`

`\sf x^6y^6z^(18) * x^4y^6z^8 * 81x^(12)81y^4z^8 \rightarrow 81x^(6+4+12)y^(6+6+4)z^(18+8+8) = \boxed{\sf 81x^(22)y^(16)z^(34)}`