Difference between revisions of "1990 AIME Problems/Problem 2"

Revision as of 16:40, 2 September 2019

Problem

Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$ .

Solution 1

Suppose that $52+6\sqrt{43}$ is in the form of $(a + b\sqrt{43})^2$ . FOILing yields that $52 + 6\sqrt{43} = a^2 + 43b^2 + 2ab\sqrt{43}$ . This implies that $a$ and $b$ equal one of $\pm1, \pm3$ . The possible sets are $(3,1)$ and $(-3,-1)$ ; the latter can be discarded since the square root must be positive. This means that $52 + 6\sqrt{43} = (\sqrt{43} + 3)^2$ . Repeating this for $52-6\sqrt{43}$ , the only feasible possibility is $(\sqrt{43} - 3)^2$ .

Rewriting, we get $(\sqrt{43} + 3)^3 - (\sqrt{43} - 3)^3$ . Using the difference of cubes, we get that $[\sqrt{43} + 3\ - \sqrt{43} + 3]\ [(43 + 6\sqrt{43} + 9) + (43 - 9) + (43 - 6\sqrt{43} + 9)]$ $= (6)(3 \cdot 43 + 9) = \boxed{828}$ .

Solution 2

The $3/2$ power is quite irritating to work with so we look for a way to eliminate that. Notice that squaring the expression will accomplish that. Let $S$ be the sum of the given expression. $S^2= ((52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2})^2$ $S^2 = (52+6\sqrt{43})^{3} + (52-6\sqrt{43})^{3} - 2((52+6\sqrt{43})(52-6\sqrt{43}))^{3/2}$ After doing the arithmetic (note that the first two terms will have some cancellation and that the last term will simplify quickly using difference of squares), we arrive at $S^2 = 685584$ which gives $S=\boxed{828}$ .

Solution 3

Factor as a difference of cubes. $\left[\left(52+6\sqrt{43}\right)^{\frac{1}{2}}-\left(52-6\sqrt{43}\right)^{\frac{1}{2}}\right]\left[\left(\left(\left(52+6\sqrt{43}\right)^{\frac{1}{2}}\right)^2+\left(52+6\sqrt{43}\right)^{\frac{1}{2}}\left(52-6\sqrt{43}\right)^{\frac{1}{2}}+\left(\left(52-6\sqrt{43}\right)^{\frac{1}{2}}\right)^2\right)\right] =$ $\left[\left(52+6\sqrt{43}\right)^{\frac{1}{2}}-\left(52-6\sqrt{43}\right)^{\frac{1}{2}}\right]\left[104+\left(52^2-\left(36\right)\left(43\right)\right)^{\frac{1}{2}}\right] =$ $\left[\left(52+6\sqrt{43}\right)^{\frac{1}{2}}-\left(52-6\sqrt{43}\right)^{\frac{1}{2}}\right]\left[104+34\right].$ We can simplify the left factor as follows. $\left(52+6\sqrt{43}\right)^{\frac{1}{2}}-\left(52-6\sqrt{43}\right)^{\frac{1}{2}} = x$ $104-2\left(52+6\sqrt{43}\right)^{\frac{1}{2}}\left(52-6\sqrt{43}\right)^{\frac{1}{2}} = x^2$ $104-68 = x^2$ $36 = x^2.$ Since $\left(52+6\sqrt{43}\right)^{\frac{1}{2}} > \left(52-6\sqrt{43}\right)^{\frac{1}{2}}$ , we know that $x=6$ , so our final answer is $(6)(138) = \boxed{828}$ .

Solution 4

Let $x=52+6\sqrt{43}$ , $y=52-6\sqrt{43}$ . Similarly to solution 2, we let $S=x^{\frac{3}{2}}+y^{\frac{3}{2}}$ $\begin{align*} S^2&=(x^{\frac{3}{2}}+y^{\frac{3}{2}})^2\\ &=x^3+y^3+2x^{\frac{3}{2}}y^{\frac{3}{2}} \end{align*}$ The expression can be simplified as follow $$ (Error compiling LaTeX. Unknown error_msg)\begin{align*} S^2&=x^3+y^3+2x^{\frac{3}{2}}y^{\frac{3}{2}}\\ &=(x+y)(x^2-xy+y^2)+2(xy)^{\frac{3}{2}}\\ &=(x+y)((x+y)^2-xy)+2\sqrt{xy}^3\\ &=104(104^2-34)+2\cdot34^3\\ &=

@@ Line 27: / Line 27: @@
 <cmath>36 = x^2.</cmath>
 Since <math>\left(52+6\sqrt{43}\right)^{\frac{1}{2}} > \left(52-6\sqrt{43}\right)^{\frac{1}{2}}</math>, we know that <math>x=6</math>, so our final answer is <math>(6)(138) = \boxed{828}</math>.
 == Solution 4 ==
+Let <math>x=52+6\sqrt{43}</math>, <math>y=52-6\sqrt{43}</math>. Similarly to solution 2, we let
+<cmath>S=x^{\frac{3}{2}}+y^{\frac{3}{2}}</cmath>
+<cmath>\begin{align*}
+S^2&=(x^{\frac{3}{2}}+y^{\frac{3}{2}})^2\\
+&=x^3+y^3+2x^{\frac{3}{2}}y^{\frac{3}{2}}
+\end{align*}</cmath>
+The expression can be simplified as follow
+<math></math>\begin{align*}
+S^2&=x^3+y^3+2x^{\frac{3}{2}}y^{\frac{3}{2}}\\
+&=(x+y)(x^2-xy+y^2)+2(xy)^{\frac{3}{2}}\\
+&=(x+y)((x+y)^2-xy)+2\sqrt{xy}^3\\
+&=104(104^2-34)+2\cdot34^3\\
+&=
 == See also ==

1990 AIME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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