OK. Quite a bit of work for 5 points, but here goes:
Picture on the left:
When square roots need to be multiplied, just go ahead and
mash everything together.
The product of √one thing) times (√another thing
is just √(one thing times the other thing) .
So √(3x²y³) · 2 · √(72x³y⁴) is just
2 · √(3x²y³ · 72x³y⁴)
= 2 · √(216 x⁵ y⁷)
That looks pretty complicated. But if you break it up into things that are
perfect squares, then you can pull those out of the square root.
2 · √(216 x⁵ y⁷)
= 2 · √(6·36 · x⁴·x · y⁶·y)
The bold things are perfect squares, so take their square roots
and write those outside of the square root sign.
= 2 · √(6·36 · x⁴·x · y⁶·y)
= 2 · 6 · x² · y³ · √(6 · x · y)
= 12 x² y³ √6xy .
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Middle picture:
How do you take the 8th root of something raised to a power ?
Just divide the power by 8 .
⁸√(p²⁰· q²⁹ · r¹¹)
= ⁸√(p¹⁶·p⁴ · q²⁴·q⁵ · r⁸·r³)
= p² q³ r ⁸√p⁴ q⁵ r³
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Picture on the right:
-3i (6 + 2i)
= (-3i)·(6) + (-3i)·(2i)
= -18i + -6i²
If ' i ' = √-1 , then i² = -1 .
-18i + -6i²
= -18i + -6(-1)
= -18i + 6 .
Make it a little bit neater. Factor out the ' 6 ':
-18i + 6
= 6·(-3i + 1) .
