Which is equivalent to RootIndex 4 StartRoot 9 EndRoot Superscript one-half x?

Mathematics

1 answer:

nata0808 [166]3 years ago

8 0

Answer:

${9}^ {\frac{1}{8}x}$

Step-by-step explanation:

We want to find an equivalent expression for

$(\sqrt[4]{9})^{ \frac{1}{2}x}$

To find an equivalent expression, we need to apply the following property of exponents:

${a}^{ \frac{m}{n}}=( \sqrt[n]{ {a}} )^{m}$

We let a=9, n=4 and m=½x

Then :

${9}^{ \frac{ \frac{1}{2}x}{4}}=( \sqrt[4]{ {9}} )^{ \frac{1}{2}x}$

Simplify the left hand side to get:

${9}^ {\frac{1}{8}x} =( \sqrt[4]{ {9}} )^{ \frac{1}{2}x}$

Therefore the correct answer is:

${9}^ {\frac{1}{8}x}$

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Two cars are driving towards an intersection from perpendicular directions.

FrozenT [24]

For a better understanding of the solution provided here please find the diagram in the attached file.

In the diagram, A is the intersection. B is the position of the first car and C is the position of the second car. As can be clearly seen, as per the directions given in the question, the cars and the intersection make a right triangle.

The distance between the first car and the intersection is $x$ and the distance between the second car and the intersection is $y$ . The distance between the two cars is depicted by $s$ . As we can see, $s$ is the hypotenuse of the right triangle. At the given instance the distances are 8 meters and 6 meters respectively. Thus, by the Pythagorean Theorem the hypotenuse will be:

$s=\sqrt{8^2+6^2}= \sqrt{100}=10$ ...............(Equation 1)

Now, we know that at any instant the Pythagorean Theorem holds and so we will have, in general:

$s^2=x^2+y^2$

Now, implicitly differentiating the above formula with respect to time, we get:

$2s\frac{ds}{dt}=2x \frac{dx}{dt}+2y \frac{dy}{dt}$

This can be further simplified by dividing both the sides by the common factor 2 as:

$s\frac{ds}{st}=x \frac{dx}{dy}+y\frac{dy}{dt}$ .................(Equation 2)

As we can see from the questions and the diagram, $\frac{dx}{dt}=-2 m/s$ and $\frac{dy}{dt}=-9 m/s$ . The negative sign is there because the distances y and x are reducing as the cars approach the intersection.