Question

Please help me with these?!

Answer 1

1. We have the formula for the volume of sphere: $V= \frac{4 \pi }{3}r^3$
where 
$V$ is the volume
$r$ is the radius 
We know from our problem that the volume of our spherical balloon is $100in^3$, so $V=100in^3$. Lets replace that value in our formula and solve for $r$:
$100in^3= \frac{4 \pi }{3}r^3$
$3(100in^3)=4 \pi r^3$
$300in^3=4 \pi r^3$
$r^3= \frac{300in^3}{4 \pi }$
$r= \sqrt[3]{\frac{300in^3}{4 \pi } }$
$r=2.879in$

We can conclude that the radius of our spherical balloon is approximately 3 inches long. Therefore, the correct answer is A. 3 in.

2. Lets simplify each one of the expressions first:
F. $\sqrt{x^2}$ since the radicand is elevated to the same number as the index of the radical, we can cancel the radical and the exponent:
$\sqrt{x^2} =x$
Since $x\ \textgreater \ 0$, this expression is equal to $x$.
G. $\frac{1}{2} \sqrt[3]{8x^3}$ 8 can be expressed as $8=2*2*2=2^3$, so we can rewrite our radicand:
$\frac{1}{2} \sqrt[3]{8x^3} =\frac{1}{2} \sqrt[3]{2^3x^3} = \frac{1}{2} \sqrt[3]{(2x)^3}$
Since the radicand is elevated to the same number as the index of the radical, we can cancel the radical and the exponent:
$\frac{1}{2} \sqrt[3]{(2x)^3}= \frac{1}{2} (2x)$
Now, we can cancel the 2 in the denominator with the one in the numerator:
$\frac{1}{2} (2x)=x$
Since $x\ \textgreater \ 0$, this expression is equal to $x$.
H. $\sqrt[3]{-x^3}$ The radicand is elevated to the same number as the index of the radical, so we can cancel the radical and the exponent:
$\sqrt[3]{-x^3}=-x$
Since $x\ \textgreater \ 0$, this expression is NOT equal to $x$.

We can conclude that the correct answer is H. $\sqrt[3]{-x^3}$.

3. The fourth root of $- \frac{16}{81}$ is $\sqrt[4]{ -\frac{16}{81} }$. Remember that negative numbers don't have real even roots since a number raised to an even exponent  is either positive or 0. Since 4 is even and $- \frac{16}{81}$ is negative, we can conclude that $- \frac{16}{81}$ has not a real fourth root. 

The correct answer is D. no real root found.

4. This time we are going to take a different approach.  We are going to simplify $\sqrt{a^2(x+a^2)}$ first, and then, we are going to compare the result with our given options:
$\sqrt{a^2(x+a^2)}$
Lets apply the product rule for a radical $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$:
$\sqrt{a^2(x+a^2)} = \sqrt{a^2} \sqrt{x+a^2}$
Notice that in our first product the radicand is raised to the same number as the index, so we can cancel the radical and the exponent:
$\sqrt{a^2} \sqrt{x+a^2}=a \sqrt{x+a^2}$

We can conclude that the correct answer is F. $a \sqrt{x+a^2}$

5. Just like before, we are going to simplify $\sqrt{4x^2y^4}$ firts, and then, we are going to compare the result with our given options. To simplify our radical expression we are going to use some laws of radicals:
$\sqrt{4x^2y^4}$
Applying product rule for a radical $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$:
$\sqrt{4x^2y^4} = \sqrt{4} \sqrt{x^2} \sqrt{y^4}$
Notice that $\sqrt{4} =2$, so:
$\sqrt{4} \sqrt{x^2} \sqrt{y^4} =2 \sqrt{x^2} \sqrt{y^4}$
Notice that $y^4=(y^2)^2$, so we can rewrite our expression:
$2 \sqrt{x^2} \sqrt{y^4}=2 \sqrt{x^2} \sqrt{(y^2)^2}$
Applying the radical rule $\sqrt[n]{a^n} =a$:
$2 \sqrt{x^2} \sqrt{(y^2)^2} =2xy^2$

We can conclude that the correct answer is A. $2xy^2$.

5. Lets check our statements:
F. $x$ is always greater than $\sqrt{x}$.
If $x=0$, $\sqrt{x} =0$. Therefore, this statement is FALSE.
G. $x= \sqrt{x}$ only if $x=0$. For a number bigger than zero the square root of that number will be always less than the number; therefore, this statement is TRUE.
H. Using a calculator we can check that $\sqrt{ \frac{1}{2} } =0.7071$ and $\frac{1}{2} =0.5$. Since $0.7071\ \textgreater \ 0.5$, we can conclude that $\sqrt{ \frac{1}{2} } \ \textgreater \ \frac{1}{2}$.
We can conclude that this statement is TRUE.
J. $x^4$ is always bigger than $x^2$.
If $x= \frac{a}{b}$ and $b\ \textgreater \ a$, $x^4$ will always be smaller than $x^2$. Therefore, we can conclude that this statement is FALSE.

6. We know that the formula for the volume of cube is $V=s^3$
where
$V$ is the volume 
$s$ is the side 
We know for our problem that a cubical storage bin has a volume of 5832 cubic inches, so $V=5832in^3$. Lets replace that value in our formula and solve for $s$:
$5832in^3=s^3$
$s^3=5832in^3$
$s= \sqrt[3]{5832in^3}$
$s=18in$

We can conclude that the length of the side of the cubical storage bin is 18 inches.