Question

Simplify the following expressions to their simplest form. Again, you must show all your working.

a) (x5y2)3 x−3y4 ÷ x3y2 (x−2y3)3
b) 2 log3(xy2) −log3(x2y2) + log9(xy)

Answer 1

To simplify the expression
`(x^5y^2)^3 * x^(-3)y^4 / x^3y^2 * (x^(-2)y^3)^3` a power and divide terms with the same base. The simplified expression is x^18 * y^9. For
`2 log3(xy^2) - log3(x^2y^2) + log9(xy)`fy and obtain
`log3(9x^3y^5).`

Step-by-step explanation:

To simplify the expression
`(x^5y^2)^3 * x^(-3)y^4 / x^3y^2 * (x^(-2)y^3)^3`ng a power to a power. In this case, raise each term inside the parentheses to the power outside the parentheses. This gives us
`x^15y^6 * x^(-3)y^4 / x^3y^2 * x^(-6)y^9`ing the terms by the same base. Divide x^15 by
`x^3, y^6 / y^2, y^9 / y^4, and x^(-3) / x^(-6)`when dividing the same base. Simplifying further, we get
`x^(15-3-(-6)) * y^(6-2-4+9)`
`x^18 * y^9`

For the expression
`2 log3(xy^2) - log3(x^2y^2) + log9(xy)` simplify. According to the property that the logarithm of a product is the sum of the logarithms of the individual factors, we can rewrite the expression as
`log3((xy^2)^2) + log9(xy)`the first term inside the logarithm, which gives us
`log3(x^2y^4) + log9(xy)`using the property that the logarithm of a sum is the sum of the logarithms, obtaining
`log3(x^2y^4 * 9xy)` ression inside the logarithm, yielding
`log3(9x^3y^5).`