Solve Sqrt. ( Sqrt 49 - Sqrt 48 ) | Math Olympiad Problem

A wonderful Math Olympiad questions. Learn the easy method to solve square root related problems.

This video tells step wise how to solve a square root. If you like the video, please like & subscribe the channel.

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[tcmxiyw]

In sqrt(x^2), the functions do not “cancel”! sqrt(x^2)=|x|. By removing the absolute value, you are assuming that x is non-negative. I know this may seem like nitpicking, but bad habits started early are hard to break. Students will get into trouble by forgetting about the absolute value when from the context it is known that x is negative. It’s easy to handle this situation by writing sqrt(x^2)=|x| and then remove the absolute value by analyzing the sign of x. sqrt(7^2)=|7|=7, but sqrt((-7)^2)=|-7|=-(-7)=7. In the latter case to “cancel” is incorrect. My students have run into difficulties when using “cancellation of functions” (e.g. integrate sqrt(f(x)^2) over an interval on which f(x) has both positive and negative values). I know this problem will come up so at the beginning of class so I ask my students “what is the value of sqrt(x^2)?” On the first day the answer is “x” almost unanimously, sometimes “plus or minus x”; by the end of the week everybody answers “absolute vale of x”. Problems like the one above are easily handled when the correct simplification sqrt(x^2)=|x| is used. Writing sqrt(7^2)=|7|=7 may seem a little silly, but it’s the beginning of developing a good habit. “Canceling” functions develops into a bad habit that leads to incorrect calculations.

[StevenTorrey]

Unfortunately, your conclusion was blocked by projected videos.

[jefflittle]

Sorry, just no. The square root of 3 is **not** 1.732. Square root of 3 is the simplified form.

[meurdesoifphilippe]

Wrong.
Correct answer is 2-√3, not 0.268 you ended with and proudly put in a box (this is only an approximation). You should have put in the final box the exact value, and written the numerical approximation with an approx. symbol or =0.267... (with dots indicating that it's the beginning of the decimal writing).

[jim]

√√49 - √48) = √7 - 2√12). Shortcut available? 4 + 3 = 7 and 4 x 3 = 12. Yep, shortcut available. Automatic: √4 - √3. Simplifies: 2 - √3.

[Nikioko]

If you split 7 wrong, you get:
√(7 - 4√3)
= √(3 - 4√3 + 4)
= √(√3² - 2 ⋅ √3 ⋅ 2 + 2²)
= √(√3 - 2)²
= √(- (2 - √3))²
= i (2 - √3).
Which is of course wrong. So, make that always a > b in a² - 2ab + b². Otherwise, you get (a - b) < 0 and thus (a - b)² being complex.

[MarcelCox]

Those nested square root problems are very common. It always astonishes me who people reinvent the wheel instead of once proving the general formula and then use it:
√(a±2√b) = √c ±√d (1)

Where c and d are the solutions of the equation (with c>d for the substraction case):

x²-ax+b=0

You can easily derive this by squaring both sides of equation (1)

In the case of this problem, the original foruma can be simplified to:


√(7±2√12)
The equation becomes:
x²-7x+12=0

And the 2 roots are 4 and 3.

[dalenassar]

Have you tried the "sum" or "difference" of two cubes? This was the first thing that came to mind at the start.

[davidbrown]

Lovely Sir. Thank you so much for this, which has induced me to subscribe.

[dont-want-no-wrench]

never sure what the point is of this kind of problem, other than symbol manipulation. no practical problem would give such a neat setup.

[albertofernandez]

Pero qué ejercicio tan bonito señor profesor

[Nikioko]

2:56: This is WRONG! √3 is an irrational number and therefore NOT equal to 1, 732. It is just APPROXIMATELY 1, 732. SO, instead of an equal sign (=) you have to write an approximation sign (≈).

[florisv]

When the problem explicitly states that no calculator is allowed, it's a sign that you should not offer an approximation. Here you offer one in three (why three?) decimals. That's a wrong answer.

[yoshinaokobayashi]

- 2sqrt(12))= sqrt( (2-sqrt(3))^2)=2-sqrt(3). Fin.

[dvgvhut]

P=given formula
P^2=7-2sqrt12
=(2-sqrt3)^2
P=2-sqrt3

[umbranocturna]

i didnt get past factoring the 4^2 out of the second root, didnt know what to do with the sqrt of 3...

[steveodonnell]

You go step by step by step to explain sqrt(49) = 7 and sqrt(48) = 4 sqrt(3) but then you leap from sqrt(7 - 4 sqrt (3)) = sqrt ( 4 + 3 - 4(sqrt (3)) = 2 - sqrt (3) ... where did that come from?

[angelmendez-rivera]

sqrt(sqrt(49) – sqrt(48)) = sqrt(7 – sqrt(48)) = sqrt(7 – 4•sqrt(3)) = sqrt(2^2 + 3 – 2•2•sqrt(3)) = sqrt((2 – sqrt(3))^2) = |2 – sqrt(3)| = |sqrt(4) – sqrt(3)| = sqrt(4) – sqrt(3) = 2 – sqrt(3) = 1/(2 + sqrt(3)).

[TrussttN]

Just by observation, I said 0.3. Close enough for government work 😂

[walterwen]

Math Olympid: √(√49 – √48) = ?
√49 – √48 = 7 – 4√3 = 4 – 2(2)(√3) + 3 = (2 – √3)^2
√(√49 – √48) = √[(2 – √3)^2] = ± (2 – √3) = ± 0.268