MHB Can the Algebraic Expression $8-2\sqrt7$ be Simplified to a Perfect Square?

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The discussion centers on determining whether the algebraic expression $8-2\sqrt7$ can be simplified to a perfect square. It is shown that $8-2\sqrt7$ can be expressed as $(\sqrt7-1)^2$ by solving the equations derived from the expansion of $(a+b\sqrt7)^2$. The values of $a$ and $b$ are found to be $-1$ and $1$, respectively. The simplification confirms that $8-2\sqrt7$ is indeed a perfect square. The conversation also touches on the implications of taking the square root of both sides, questioning the complexity of the algebra involved.
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See picture. Show that the RHS = LHS without using a calculator.

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Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2$$

$$a^2+7b^2=8$$

$$2ab=-2$$

$$\implies a=-1,b=1$$

Hence $8-2\sqrt7=(\sqrt7-1)^2$

$$\sqrt7-\sqrt{8-2\sqrt7}=\sqrt7-(\sqrt7-1)=\sqrt7-\sqrt7+1=1$$
 
greg1313 said:
Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b$$
Check the last term.
[math](a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2[/math]

-Dan
 
topsquark said:
Check the last term.

Post corrected. Thanks Dan.
 
greg1313 said:
Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2$$

$$a^2+7b^2=8$$

$$2ab=-2$$

$$\implies a=-1,b=1$$

Hence $8-2\sqrt7=(\sqrt7-1)^2$

$$\sqrt7-\sqrt{8-2\sqrt7}=\sqrt7-(\sqrt7-1)=\sqrt7-\sqrt7+1=1$$

What would the algebra look like if I take the square root on both sides? Is it tedious?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

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