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Good question, but you should also indicate what you tried and what your best thinking is on the situation.

In any case this is a good exercise in the application of the laws of exponents.

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I believe the expression is

³ sqrt (3xy^2)/ (81x^4y^2).

The parenthese around the denominator are essential; otherwise the only factor in the denominator is 81 and the rest of the expression would be multiplied by x^2 y^2.

The numerator is the cube root of 3 x y^2.

The cube root is the 1/3 power, so the numerator can be expressed as

(3 x y^2) ^ (1/3)

Using the laws of exponents we can write this as

3^(1/3) x^(1/3) y^(2/3).

The original expression therefore becomes

3^(1/3) x^(1/3) y^(2/3) / (81 x^4 y^2).

This can be written as

(3^(1/3) / 81) * (x^(1/3) / x^4) * (y^(2/3) / y^2),

81 = 3^4, so 3^(1/3) / 81 = 3^(1/3) / 3^4 = 3^(1/3 - 4) = 3^(1/3 - 12/3) = 3^(-11/3).

The other terms can be simplified in a similar manner, so that we end up with

3^(-11/3) * x^(-11/3) * y^(-4/3).

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Now it isn't clear from your question in what form the result is to be written. We will find out and address this in class.

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I'll work out the similar problem

(x-3)(2x+4) = (3x-5) * x

If you apply the distributive law you get

x^2 + 2 x - 12 = 3x^2 - 5 x

You should at this point recognize this as a quadratic equation, which needs to be put into the form

a x^2 + b x + c = 0.

This is easily done. Subtracting the right-hand side from both sides and multiplying by -1 we obtain

2 x^2 + 7 x - 12 = 0.

If this factors it's going to take a little work, so we check out the discriminant:

The discriminant (the part under the square root in the quadratic formula) is

discriminant = 7^2 - 4 * 2 * (-12) = 49 + 96 = 145.

Since 144 is the square of 12 and 169 the square of 13, we conclude that the discriminant is an irrational number between 12 and 13, so factoring will not be a possibility (the factors would end up being irrational, so we would never be able to find them).

We therefore solve this equation by applying the quadratic formula. I believe one of the solutions will be

x = -7/4 + sqrt(145) / 4.

You should verify this and find the other solution as well.

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