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3 years ago

What is the monomial if a square of a monomial is: 1/4 a2n

Mathematics

2 answers:

lilavasa [31]3 years ago

7 0

Answer:

$\frac{1}{2} a^n$

Step-by-step explanation:

Square of the monomial is $\frac{1}{4}a^{2n}$

To get the monomial we take square root

Lets take square root for each term

$\sqrt{\frac{1}{4}a^{2n}}$

$\sqrt{\frac{1}{4}}=\fract{1}{2}$ because square root of 4 is 2

$\sqrt{a^{2n}}=(a^{2n})^\frac{1}{2}$

2/2 is 1 so, its a^n

Required monomial is $\frac{1}{2} a^n$

kipiarov [429]3 years ago

7 0

Answer:

$\frac{1}{2}a^n$

Step-by-step explanation:

Givens:

Square of a monomial: $\frac{1}{4} a^{2n}$ .

The problem is asking for the monomial, and the given is squared. What we have to do is to extract that square from expression, and that it's done applying a squared root, because that's the opposite operation of a squared power:

$\sqrt{\frac{1}{4} a^{2n}}$

So, here we have to find the squared root of $\frac{1}{4}$ and $a^{2n}$ , applying the root to each factor:

$\sqrt{\frac{1}{4}} \sqrt{a^{2n}}$

So, we know that a root can be expressed as a fractional exponent, and also the root of a fraction is the root of each part of the fraction:

$\frac{\sqrt{1}}{\sqrt{4}} (a^{2n})^{\frac{1}{2} }$

Now, we solve:

$\frac{1}{2}a^{\frac{2n}{2}}= \frac{1}{2}a^n$

Therefore, the monomial expression is $\frac{1}{2}a^n$

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A population of bees is decreasing. The population in a particular region this year is 1,250. After 1 year, it is estimated tha

8_murik_8 [283]

To model this situation, we are going to use the decay formula: $A=Pe^{rt}$
where
$A$ is the final pupolation
$P$ is the initial population
$e$ is the Euler's constant
$r$ is the decay rate
$t$ is the time in years

A. We know for our problem that the initial population is 1,250, so $P=1250$ ; we also know that after a year the population is 1000, so $A=1000$ and $t=1$ . Lets replace those values in our formula to find $r$ :
$A=Pe^{rt}$
$1000=1250e^{r}$
$e^{r}= \frac{1000}{1250}$
$e^{r}= \frac{4}{5}$
$ln(e^{r})=ln( \frac{4}{5} )$
$r=ln( \frac{4}{5} )$
$r=-02231$

Now that we have $r$ , we can write a function to model this scenario:
$A(t)=1250e^{-0.2231t}$ .

B. Here we are going to use a graphing utility to graph the function we derived in the previous point. Please check the attached image.

C.
- The function is decreasing
- The function doe snot have a x-intercept
- The function has a y-intercept at (0,1250)
- Since the function is decaying, it will have a maximum at t=0:
$A(0)=1250e^{(-0.2231)(0)$
$A_{0}=1250e^{0}$
$A_{0}=1250$
- Over the interval [0,10], the function will have a minimum at t=10:
$A(10)=1250e^{(-0.2231)(10)$
$A_{10}=134.28$

D. To find the rate of change of the function over the interval [0,10], we are going to use the formula: $m= \frac{A(0)-A(10)}{10-0}$
where
$m$ is the rate of change
$A(10)$ is the function evaluated at 10
$A(0)$ is the function evaluated at 0
We know from previous calculations that $A(10)=134.28$ and $A(0)=1250$ , so lets replace those values in our formula to find $m$ :
$m= \frac{134.28-1250}{10-0}$
$m= \frac{-1115.72}{10}$
$m=-111.572$
We can conclude that the rate of change of the function over the interval [0,10] is -111.572.