Question

all natural numbers a so that expression <br /><br />$\sqrt{a + 64 \div a - 64}$<br />is also a natural number​

Answer 1

Answer: Only possible value of a is 1 .

Step-by-step explanation:

$\sqrt{a + 64 / a- 64 }$

By Bodmas rule , we have :-

=> $\sqrt{a + \frac{64}{a} - 64 }$

=> $\sqrt{\frac{a^2 + 64 - 64a}{a} }$

Since the expression is a rational number , then the expression $\frac{a^2+64 - 64a}{a}$ must be a perfect square .

Suppose $\frac{a^2 + 64 - 64a}{a} = k^2$ for some k .

Then a² - 64a + 64 = ak²

Notice that by using a bit of logic , the RHS side is divisible by a , so the LHS side must also be divisible by a .

This implies that a divides 64 .

So the possible values of a are (1,2,4,8,16,32,64) .

Note that only for a = 1 , we get √a + 64 ÷ a - 64 is a natural no. , since for every other value of a , we have the square root of a negative number , which is imaginary (complex nos.)

So the only value of a is 1 .

Hope this helps you .

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