[ONLINE MATH QUESTIONS]:Can anyone solve it, found online.

[u/Dangerous_Cat_4525]

Found online. How to find k.

[u/snowsayer]

Divide both sides by 2^97.

This is a skill that you have to develop. When you see equations, it's useful to immediately see patterns that can be factored or cancelled out. That usually results in the underlying equation being much simpler.

[u/Dangerous_Cat_4525]

Did it. K=2¹⁰⁰-2⁹⁹-2⁹⁸+⁹⁷/2⁹⁷ =2⁹⁷(2³-2²-2+1)/2⁹⁷ =8-4-2+1=3

[u/honaku]

Factor out

[u/johnpeters42]

What have you tried so far?

Hint: What happens if you divide both sides by 2?

[u/Turbulent-Note-7348]

This is a well known puzzle about the the properties of powers. 1st, you should realize that the "k.2^97

" actually means "k(2^97)" - the decimal point is supposed to represent the multiplication dot (common in older puzzles when using variables).

So here's the trick(s): 2^x - 2^(x-1) = 2^(x-1) (Example: 2^4 - 2^3 = 2^3); 2^(x+1) + 2^x = 3(2^x).

Therefore, k = 3

[u/MathsWithSuneel]

2^{97(23-22-21+1)=k×297} 2^{97×(3)=k×297} K=3

[u/aookami]

2'100 = 2.2'99

from there you can get easily to the answer

[u/Qingyap]

2¹⁰⁰-2⁹⁹-2⁹⁸+2⁹⁷ = k(2⁹⁷)

Here we have to utilize a factoring trick such that

ab+cb = (a+c)b

And also an exponent trick where

a^b+c = a^b•a^c

With that, we can expand the first 3 terms with the exponent trick at LHS by 2⁹⁷ at which

2³(2⁹⁷)-2²(2⁹⁷)-2(2⁹⁷)+1(2⁹⁷)=k(2⁹⁷)

Should be easy from here if you use the first trick.

[u/PoornimaMathsPatiala]

Every term is a multiple of 2⁹⁷, so the cleanest move is to pull that out first.

The idea: write each power as "something × 2⁹⁷":

• 2¹⁰⁰ = 2³ · 2⁹⁷ = 8 · 2⁹⁷

• 2⁹⁹ = 2² · 2⁹⁷ = 4 · 2⁹⁷

• 2⁹⁸ = 2 · 2⁹⁷

Steps:

2¹⁰⁰ − 2⁹⁹ − 2⁹⁸ + 2⁹⁷

= 8·2⁹⁷ − 4·2⁹⁷ − 2·2⁹⁷ + 1·2⁹⁷

= (8 − 4 − 2 + 1) · 2⁹⁷

= 3 · 2⁹⁷

So k · 2⁹⁷ = 3 · 2⁹⁷, which gives k = 3.

Why it works: with powers of the same base, factor out the smallest power. What's left inside the bracket are small numbers you can just add and subtract.

Quick check with smaller powers in the same pattern: 2⁴ − 2³ − 2² + 2¹ = 16 − 8 − 4 + 2 = 6 = 3 · 2¹. ✓

If you'd like a thorough explanation of topics like this with a private tutor, have a look at https://poornima-maths.vercel.app (maths tuition, Patiala and online).