To simplify this expression, you need to follow the rules for manipulation of exponents.
Let's first simplify the fraction inside the bracket:
The given expression is (6x^6 * y^2)/(18x^3 * y^-7)
Here, we can simplify this by separately treating the two parts:
1) x's with their exponents.
2) y's with their exponents.
1) Simplify the x terms:
The numerator part of the fraction has a term with x, which is x^6. The denominator contains x^3.
When you divide with the same base (x), you subtract the exponents.
So, x^6 divided by x^3 simplifies to x^(6-3) = x^3.
2) Simplify the y terms:
In the numerator we have y^2 and in the denominator we have y^-7.
y^2 divided by y^-7 simplifies to y^(2-(-7)) = y^9.
Remember, subtracting a negative sign is equivalent to addition.
Next, simplify coefficient:
The numerator of our original expression has a coefficient 6 and the denominator has a coefficient of 18.
Dividing 6 by 18 simplifies to 1/3.
After simplifying, we get the fraction as x^3*y^9/3.
Next, we have the entire fraction to the power of -4.
Power to a power multiplies. Hence, each of the exponents in our fraction will be multiplied by -4:
(x^3*y^9/3)^(-4)
This simplifies to:
(x^(-12) * y^(-36))/81
Because all exponents should be positive, we take the inverse of the base. A base with a negative exponent takes the reciprocal form.
This will be our final result:
81/(x^12 * y^36)