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# how should i verify my solution by using decimal approximations?

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expression: $${4\over 3-2\sqrt{2}}$$

answer: $${12 +8 \sqrt{2}}$$

and please show step by step how you did it ^.^

Oct 2, 2017

#1
+2

Simplify the following:
4/(3 - 2 sqrt(2))

Multiply numerator and denominator of 4/(3 - 2 sqrt(2)) by 2 sqrt(2) + 3:
(4 (2 sqrt(2) + 3))/((3 - 2 sqrt(2)) (2 sqrt(2) + 3))

(3 - 2 sqrt(2)) (2 sqrt(2) + 3) = 3×3 + 3×2 sqrt(2) - 2 sqrt(2)×3 - 2 sqrt(2)×2 sqrt(2) = 9 + 6 sqrt(2) - 6 sqrt(2) - 8 = 1:
(4 (2 sqrt(2) + 3))/(1)

(4 (2 sqrt(2) + 3))/(1) = 4 (2 sqrt(2) + 3):
4 (2 sqrt(2) + 3) =8sqrt(2) + 12

Oct 2, 2017
#2
+101046
+2

$$\sqrt2\approx 1.414213562$$

I suppose you could just use closer and closer estimations of sqrt2 and see if the answers keep getting closer...

$$\sqrt2 \approx 1.4\\ {4\over 3-2\sqrt{2}}\approx {4\over 3-2*1.4}\approx \frac{4}{0.2}\approx 20\\ 12+8\sqrt2\approx 12+8*1.4=12+8+3.2=23.2$$

$$\sqrt2 \approx 1.41\\ {4\over 3-2\sqrt{2}}\approx {4\over 3-2*1.41}\approx \frac{4}{0.18}\approx 22.2\\ 12+8\sqrt2\approx 12+8*1.41=12+11.28=23.28$$