ANSWER TO QUESTION 1
A rational number is any number that can be written in the form,[tex]\frac{a}{b}[/tex], where [tex]a[/tex] and [tex]b[/tex] are integers and [tex]b\ne 0[/tex].
We can rewrite [tex]-4=\frac{-4}{1}[/tex].
Therefore option A is a rational number.
Option B is obviously a rational number because it is already in the form [tex]\frac{a}{b}[/tex].
Option C cannot be written in the form [tex]\frac{a}{b}[/tex] because the decimal place does not repeat or recur and it does not terminate also.
Therefore [tex]4.51010910921...[/tex] is not a rational number.
As for option D, the decimal places recurs or repeats and it does not terminate. We can rewrite in the form,[tex]\frac{a}{b}[/tex].
[tex]0.987987987...=\frac{329}{333}[/tex] hence it is a rational number.
ANSWER TO QUESTION 2
Yes [tex]81[/tex] is a perfect square.
All numbers whose square roots are perfect squares are rational numbers.
If we raise [tex]9^{2}[/tex] we get [tex]81[/tex].
In order words if we take the square root of [tex]81[/tex] we get a rational number.
That is [tex]\sqrt{81} =9[/tex]
ANSWER TO QUESTION 3
Let the length from the wall to the base of the ladder be [tex]x[/tex]m.
The from Pythagoras Theorem,
[tex]l^2+12^2=13^2[/tex]
This implies that,
[tex]l^2+144=169[/tex]
We add the additive inverse of [tex]144[/tex] to both sides to obtain,
[tex]l^2=169-144[/tex]
[tex]l^2=25[/tex]
We take the positive square root of both sides to obtain,
[tex]l=\sqrt{25}[/tex]
[tex]l=5[/tex].
The correct answer is A.
ANSWER TO QUESTION 4
We wan to estimate [tex]\sqrt{52}[/tex].
The highest perfect square that can be found in [tex]52[/tex] is [tex]4[/tex].
We rewrite to obtain,
[tex]\sqrt{52} =\sqrt{4\times 13}[/tex].
We now split the square root sign to obtain,
[tex]\sqrt{52} =\sqrt{4}\times \sqrt{13}[/tex].
[tex]\sqrt{52} =2\sqrt{13}[/tex].
[tex]\sqrt{52} =2(3.60)[/tex].
[tex]\sqrt{52} \approx 7.21[/tex].
ANSWER TO QUESTION 5.
The statement, every rational number is q square root is false.
We only need at least a counterexample to show that, the above statement is false.
Let [tex]x[/tex] be any real number.
Then [tex]\sqrt{x} =\frac{a}{b}[/tex], where [tex]a[/tex] and [tex]b[/tex] are integers.
This implies that [tex]a=b\sqrt{x}[/tex]
Base on this final equation, [tex]a[/tex] can only be an integer if [tex]x[/tex] is a perfect number. Hence not every rational number is a square root because some numbers aren't perfect squares.
The correct answer is C
ANSWER TO QUESTION 6.
To find the translation vector that maps the blue rectangle on the red rectangle, we draw a vector connecting any two corresponding points as shown in the diagram.
The vector has horizontal component of [tex]6[/tex] and a vertical component of [tex]-5[/tex].
Therefore the mapping is [tex](x,y)\rightarrow(x+6,y-5)[/tex]. The correct answer is D.
ANSWER TO QUESTION 7
Figure A accurately represents the Pythagorean Theorem because
[tex]625=576+49[/tex]
This implies that
[tex]25^2=24^2+7^2[/tex]
We can see that the hypotenuse square is equal to the sum of the squares of the lengths of the two shorter legs.
Recall that, [tex]7,24,25[/tex] are Pythagorean triples.
