D01-01

Question

What is the value of  \sqrt{324} + \sqrt{289} ?

A. 31
B. 33
C. 34
D. 35
E. 36

Bunuel · Accepted Answer

Official Solution:

What is the value of  \sqrt{324} + \sqrt{289} ?

A. 31
B. 33
C. 34
D. 35
E. 36

The square root of a positive integer is either an integer or an irrational number.  This means that the square root of a positive integer cannot be a fraction, such as  \frac{1}{2}, \ \frac{3}{7}, \ \frac{19}{2}, \ \frac{1}{60} , and so on. It must be either an integer (e.g., 1, 2, 3, ...) or an irrational number (e.g.,  \sqrt{2} ,  2\sqrt{3} ,  5\sqrt{7} , ...). However, given that the answer choices provided are integers, both  \sqrt{324}  and  \sqrt{289}  must also be integers. This is because the sum of two positive irrational numbers, such as  \sqrt{2} ,  2\sqrt{3} ,  5\sqrt{7} , and so forth, will not result in an integer. This eliminates the possibility of irrational square roots in this context.
 
Next,  \sqrt{324}  is less than  \sqrt{400}  (which is 20), and must end with either 2 or 8. Given that  12^2 = 144 , we can deduce that  \sqrt{324} = 18 .
 
Consequently,  \sqrt{289}  must be less than 18 and end with either 3 or 7. Since  13^2 = 169 , we can deduce that  \sqrt{289} = 17 .
 
Therefore, the sum of  \sqrt{324}  and  \sqrt{289}  is  18 + 17 = 35 .

Answer: D

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

einstein801 · Answer

How do you know that sqrt(289) must end in 3 or 7?

Bunuel · Answer

Because only 3^2 and 7^2 end with 9.

BelisariusTirto · Answer

Sorry if this is supposed to be really intuitive or clear but for the same reason is 6 not included in the thought of process of figuring out sqrt(324) because only 2^2 and 8^2 results in 4s? Thank you for the work you do. (4, 54)

Bunuel

unicornilove

How do you know that sqrt(289) must end in 3 or 7?

Bunuel · Answer

Correct. A number ending in 6, raised to any positive integer power, will always end in 6:

6^1 = 6
6^2 = 36
6^3 = 216
...
16^2 = 256
...
and so on.

For x^2, where x is a positive integer, to end with 4, the units digit of x must be either 2 or 8.

Kanade · Answer

I like the solution - it’s helpful.

dradi123 · Answer

I like the solution - it’s helpful.

jana1202 · Answer

I like the solution - it’s helpful.

Amarinelo · Answer

I like the solution - it’s helpful. If the square root in questions isn't memorized. Use the the rule below:

Square numbers have a cyclic ending. Ex. ..1,..4,..9,...6,..5,..6,...9,...4,...1 note that they add to 10 like 1 and 9 and 4 and 6 then 5 and repeat.

Knowing that you can figure out the ending of any big number squared. or when in a Square root you can figure out the results. Ex.

We don't know the square root of 324 then we should round it, as we know the square root of 400 which is 20 so we know that the number we are looking for is between 10 and 20, and we know that our number ends in 4 so it would be either 12 or 18. 12 is 144 too low so we need 18. same idea for the other one.

AB18765 · Answer

I like the solution - it’s helpful.

D01-01

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