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Please help me with these?!

asked Nov 25, 2019 68.8k views

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Joscha · Answer 1 · 2019-12-02T08:08:06+0000

1. We have the formula for the volume of sphere:
$V= (4 \pi )/(3)r^3$
where

is the volume

is the radius
We know from our problem that the volume of our spherical balloon is
100in^3

, so

. Lets replace that value in our formula and solve for

:

$100in^3= (4 \pi )/(3)r^3$

$3(100in^3)=4 \pi r^3$

$300in^3=4 \pi r^3$

$r^3= (300in^3)/(4 \pi )$

$r= \sqrt[3]{(300in^3)/(4 \pi ) }$

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.
√(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:

√(x^2) =x

Since
$x\ \textgreater \ 0$ , this expression is equal to

.
G.
$(1)/(2) \sqrt[3]{8x^3}$ 8 can be expressed as
8=2*2*2=2^3

, so we can rewrite our radicand:

$(1)/(2) \sqrt[3]{8x^3} =(1)/(2) \sqrt[3]{2^3x^3} = (1)/(2) \sqrt[3]{(2x)^3}$
Since the radicand is elevated to the same number as the index of the radical, we can cancel the radical and the exponent:

$(1)/(2) \sqrt[3]{(2x)^3}= (1)/(2) (2x)$
Now, we can cancel the 2 in the denominator with the one in the numerator:

(1)/(2) (2x)=x

Since
$x\ \textgreater \ 0$ , this expression is equal to

.
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

.

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

3. The fourth root of
- (16)/(81)

is
$\sqrt[4]{ -(16)/(81) }$ . 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
- (16)/(81)

is negative, we can conclude that
- (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
√(a^2(x+a^2))

first, and then, we are going to compare the result with our given options:

√(a^2(x+a^2))

Lets apply the product rule for a radical
$\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$ :

√(a^2(x+a^2)) = √(a^2) √(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:

√(a^2) √(x+a^2)=a √(x+a^2)

We can conclude that the correct answer is F.
a √(x+a^2)

5. Just like before, we are going to simplify
√(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:

√(4x^2y^4)

Applying product rule for a radical
$\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$ :

√(4x^2y^4) = √(4) √(x^2) √(y^4)

Notice that
√(4) =2

, so:

Notice that
y^4=(y^2)^2

, so we can rewrite our expression:

2 √(x^2) √(y^4)=2 √(x^2) √((y^2)^2)

Applying the radical rule
$\sqrt[n]{a^n} =a$ :

2 √(x^2) √((y^2)^2) =2xy^2

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

.

5. Lets check our statements:
F.

is always greater than
√(x)

.
If

,

. Therefore, this statement is FALSE.
G.
x= √(x)

only if

. 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
$\sqrt{ (1)/(2) } =0.7071$ and
(1)/(2) =0.5

. Since
$0.7071\ \textgreater \ 0.5$ , we can conclude that
$\sqrt{ (1)/(2) } \ \textgreater \ (1)/(2)$ .
We can conclude that this statement is TRUE.
J.
x^4