Mathematics: The fourth roots of 1 are x^4=1 or x^4-1=0. how many fourth roots of 1 are there?

The fourth roots of 1 are x^4=1 or x^4-1=0. how - id1414101578568, pickelswolf3036

Ответ:

audriexpress

01.08.2019

Factoring difference of squares: $a^2-b^2 = (a+b)(a-b)$ $\begin{aligned} x^4-1 & = 0 \\ (x^2)^2 - (1)^2 & = 0 \\ (x^2+1)(x^2-1) & = 0 \\ (x^2+1)(x+1)(x-1) & = 0 \end{aligned}$ we see that we have real roots of $x=-1$ , $x=1$ .but $(x^2+1)$ can be factored over the complex numbers. since $i^2 = (-1)$ , we have $\begin{aligned} x^2+1 & = x^2 - 1(-1) \\ & = x^2 - i^2 \\ & = (x+i)(x-i) \end{aligned}$ so we also have complex roots of $x=i$ , $x=-i$ .we have a total of 4 fourth roots of 1 (2 real roots, 2 complex roots).

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Ответ:

mhortin

15.03.2022

Mathematics

Answer with explanation:

Let p represents the population proportion.

From the given information, we have

$H_0: p=0.06\\\\H_a: p0.06$ , since alternative hypothesis is right tailed then the test is right tailed test.

Given : In a sample of 416 households that owned one or more vacation homes, 46 were minorities.

i.e. n= 416

Sample proportion : $\hat{p}=\dfrac{46}{416}=0.110576923077\approx0.11$

Test statistic : $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

i.e. $z=\dfrac{0.11-0.06}{\sqrt{\dfrac{0.06(1-0.06)}{416}}}$

$\Rightarrow\ z=4.29414907893\approx4.29$

P-value for right tailed test : $P(z4.29)=1-P(z\leq4.29)$

=1-0.999991=0.0000089

Since the p-value is less than the significance level (0.01), so we reject the null hypothesis.

Thus , we conclude that we have enough evidence to support the claim that that the percentage of vacation-home owners who are minorities is larger than 6 percent.

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The fourth roots of 1 are x^4=1 or x^4-1=0. how many fourth roots of 1 are there?

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