I have tried multiple ways to solve this:
$$ \left (2\sqrt 8 - 3\sqrt 5 \right)^2 $$
and my answer was way off. Can someone show me their method? Thanks.
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As a professional mathematician, I have never heard of a "surd" before. Is this an invention of the k-12 teachers? – Steven Gubkin Mar 10 '14 at 13:17
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1@StevenGubkin It is a very ancient term, which has somewhat fallen into disuse in the last seventy or eighty years. It is certainly not an invention of the k-12 teachers. – MJD Mar 10 '14 at 13:32
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Nearly identical: Simplify the surd expression. (My simplifying comment applies here as well, of course.) – Dave L. Renfro Mar 10 '14 at 21:00
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The simplest way to prevent any careless mistakes is to apply the identity
$$(a - b)^2 = a^2 + b^2 - 2ab$$
Substitute $a = 2\sqrt{8}$ and $b = 3\sqrt{5}$. Then we have $$a^2 = (2\sqrt8)^2 = 4\cdot8 = 32$$ and $$b^2 = (3\sqrt{5})^2 = 9\cdot5 = 45$$ and $$2ab = 2\cdot2\sqrt8\cdot3\sqrt5=12\sqrt{40}$$
So we have $$(2\sqrt{8} - 3\sqrt{5})^2 = 32 + 45 - 12\sqrt{40} = 77 - 12\sqrt{40}$$
Yiyuan Lee
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You forgot to add the two -6's at the end, other than that marked as answered. Thank you for your time. – chargedparticle Mar 10 '14 at 12:59
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Do not forget to simplify $12\sqrt{40}$ to $24\sqrt{10}$ :) – Anonymous Computer Mar 10 '14 at 20:47
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Remember the formula: $$(a-b)^2=a^2-2ab+b^2$$ Apply it to your expression $(2\sqrt{8}-3\sqrt{5})^2$. But first, simplify $2\sqrt{8}$ to $4\sqrt{2}$. Now we have to find the value of: $$(4\sqrt{2}-3\sqrt{5})^2$$ Use the $(a-b)^2=a^2-2ab+b^2$ formula. $$a=4\sqrt{2}$$ $$b=3\sqrt{5}$$ $$(4\sqrt{2}-3\sqrt{5})^2=(4\sqrt{2})^2-2(4\sqrt{2})(3\sqrt{5})+(3\sqrt{5})^2$$ Remember, $(ab)^2=a^2b^2$. $$(4\sqrt{2})^2-2(4\sqrt{2})(3\sqrt{5})+(3\sqrt{5})^2=4^2(\sqrt{2})^2-2(4\sqrt{2})(3\sqrt{5})+3^2(\sqrt{5})^2$$ $$4^2(\sqrt{2})^2-2(4\sqrt{2})(3\sqrt{5})+3^2(\sqrt{5})^2=16(2)-2(4\sqrt{2})(3\sqrt{5})+9(5)$$ $$16(2)-2(4\sqrt{2})(3\sqrt{5})+9(5)=32-2(4\sqrt{2})(3\sqrt{5})+45$$ $$32-2(4\sqrt{2})(3\sqrt{5})+45=77-2(4\sqrt{2})(3\sqrt{5})$$ Now we will simplify $2(4\sqrt{2})(3\sqrt{5})$. $$77-2(4\sqrt{2})(3\sqrt{5})=77-2(4)(3)(\sqrt{2})(\sqrt{5})$$ Rememeber. $\sqrt{a}\cdot\sqrt{b}$. $$77-2(4)(3)(\sqrt{2})(\sqrt{5})=77-24\sqrt{2\cdot 5}$$ $$77-24\sqrt{2\cdot 5}=77-24\sqrt{10}$$ $$\displaystyle \boxed{\therefore \left(2\sqrt{8}-3\sqrt{5}\right)^2=77-24\sqrt{10}}$$
Anonymous Computer
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