Answer:
The answer is b. 74 and c. 80.
Step-by-step explanation:
The square root of a number is between 8 and 9 or:
[tex]\\ 8<\sqrt{x}< 9[/tex] [ 1 ] or
[tex]\\ 8\leq\sqrt{x} \leq 9[/tex] [ 2 ] (if we take 8 and 9 in the set of the solution).
Suppose we raise each member of [ 1 ] to the second power, that is:
[tex]\\8^{2} < (\sqrt{x})^{2} < 9^{2}[/tex]
We already know that [tex]\\\sqrt{x} = x^{\frac{1}{2}}[/tex].
And also that:
[tex]\\(x^{\frac{1}{2})^{2}} = x^{2*\frac{1}{2}} =x^{\frac{2}{2}} = x^{1} = x[/tex]
Then:
[tex]\\64 < x < 81[/tex]
That is, the number whose square root is between 8 and 9, is thus between 64 and 81; so, the possible answers, considering the options given, are then 74 and 80, because these numbers are between 64 and 81 (8.6 and 81.5 are not). We can represent this mathematically as follows:
[tex]\\64 < 74 < x < 80 < 81[/tex]
That is:
[tex]\\64 < 74 < 81[/tex] and
[tex]\\64 < 80 < 81[/tex]
In other words, if we obtain the root squares of each number (74 and 80), all of them are between 8 and 9:
[tex]\\\sqrt{64} <\sqrt{74} < x <\sqrt{80} <\sqrt{81}[/tex]
Or, approximately:
[tex]\\8 < 8.602 < x < 8.944 < 9[/tex]
So, 74 and 80 are the numbers whose root squares are between 8 and 9.
