MHB Can the No $4\sqrt{4-2\sqrt {3}}+\sqrt{97-56\sqrt 3}$ be an Integer?

  • Thread starter Thread starter solakis1
  • Start date Start date
  • Tags Tags
    Integer
Click For Summary
The expression \(4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}\) is analyzed to determine if it can be an integer. The first part, \(4\sqrt{4-2\sqrt{3}}\), simplifies to a rational number, while the second part, \(\sqrt{97-56\sqrt{3}}\), is examined for its integer properties. Through algebraic manipulation and simplification, it is shown that the overall expression does not yield an integer value. The conclusion is that the expression cannot be an integer, as proven through the calculations provided. Thus, the expression \(4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}\) is definitively not an integer.
solakis1
Messages
407
Reaction score
0
Can the No :$4\sqrt{4-2\sqrt {3}}+\sqrt{97-56\sqrt 3}$ be an iteger ,if yes prove it if no then prove it again
 
Mathematics news on Phys.org
integer ...

$4-2\sqrt{3} = 3 - 2\sqrt{3} +1 = (\sqrt{3} -1)^2$

$97-56\sqrt{3} = 49 - 2(28\sqrt{3}) + 48 = (7 - 4\sqrt{3})^2$

$4\sqrt{(\sqrt{3}-1)^2} + \sqrt{(7-4\sqrt{3})^2} = 4\sqrt{3}-4 + 7 -4\sqrt{3} = 3$
 
Last edited by a moderator:
very good ,excellent
 

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 »

Similar threads

MHB Integer Solutions for $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solve the Equation involving $4^{\sqrt{log_2 x}}$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Equation 2: Prove that ##x^2+2x\sqrt x+3x+2\sqrt x+1=0##
  • · Replies 4 ·
Replies
4
Views
1K
MHB Solve Equation: $\sqrt{1+\sqrt{1-x^2}}$
  • · Replies 3 ·
Replies
3
Views
1K
High School How to express ##3^{\sqrt(2)}## in terms of natural logarithms
  • · Replies 10 ·
Replies
10
Views
2K
MHB Prove Triangle Inequality: $\frac{a}{\sqrt[3]{4b^3+4c^3}}+...<2$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Proving Series Inequality: $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove Triangle Inequality: $\sqrt{2}\sin A-2\sin B+\sin C=0$
  • · Replies 2 ·
Replies
2
Views
2K
MHB Simplifying an Expression: $\sqrt{3} + \sqrt{2} / \sqrt{3} - \sqrt{2}$
  • · Replies 3 ·
Replies
3
Views
2K
Solve the given trigonometry equation
  • · Replies 2 ·
Replies
2
Views
2K