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Notice that your expression is the product of many "smaller" expressions:
81, x^12, y^8 and z^16
Therefore, in order to find what expression raised to the fourth power yields 81x^12y^8z^16, we can do this for each of the smaller expressions.
- 81: this is 3 to the 4th power. So we have: 81 = 3^4
- x^12: This is the same as (x^3)^4. Notice that I'm using the following rule:
So we have: x^12 = (x^3)^4
- y^8: We use the same rule as above to get y^8 = (y^2)^4
- z^16: Once again, we use the same rule, getting z^16 = (z^4)^4
Up to now, we've found that
81x^12y^8z^16 = 3^4*(x^3)^4*(y^2)^4*(z^4)^4
So now we use the rule:
Applying that rule, we conclude that:
3^4*(x^3)^4*(y^2)^4*(z^4)^4 = ( 3(x^3)(y^2)(z^4) )^4
Therefore, we've found that 3(x^3)(y^2)(z^4) to the 4th power is equal to your expression.
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