Alan

As follows

\((3^2\times27\times9^2)^{1/3}\rightarrow(3^2\times3\times9\times9^2)^{1/3}\rightarrow(3^3\times9^3)^{1/3}\rightarrow3\times9\rightarrow27\)

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Using imaginary numbers you could do the following

a² + 4b² = (a + 2ib)(a - 2ib)

9a² + 25b² = (3a + 5ib)(3a - 5ib)

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"Factorize by remarkable identities:49(x+1)squared minus 25"

\(49(x+1)^2-25\rightarrow(7x+7)^2-5^2\rightarrow(7x+7+5)(7x+7-5)\rightarrow(7x+12)(7x+2)\)

You need complex numbers to do this:

\((x-2-\sqrt7i)(x-2+\sqrt7i)=x^2-4x+11\)

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Plotting t*cos(t) on the x axis and t*sin(t) on the y axis does, indeed, generate a spiral.

There isn't much point in trying to express y in terms of x without the t as there are multiple values (an infinite number) of y for a given x (and vice-versa).

If you plot t*cos(t) on the y axis and t*sin(t) on the x axis you get a spiral that curves in the opposite direction.

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4- (2sinx)^2 → 4 - 4sin(x)^2 → 4(1 - sin(x)^2) →  4cos(x)^2

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See correction below.

Oops!  Thanks Guest#2, you are correct.

The volume V should be:  \(V=98\pi (y-2y^3/3+y^5/5)_1\rightarrow98\pi\times8/15\rightarrow784\pi/15\)

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"what is my score out of 100% if 18 was max and i got 10."

score as a percentage:     100*10/18 → 55.6%

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"Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 7y2, x = 7; about x = 7"