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olga_2 [115]
3 years ago
9

I need help simplifying a math problem. It is (x^2 * x^-3)^4

Mathematics
1 answer:
Mashutka [201]3 years ago
4 0
(x^2 \times x^{-3})^4 \\ \\ (x^{2-3})^4 \ / \ product \ rule \\ \\ (x^{-1})^4 \ / \ simplify \\ \\ x^{-4} \ / \ simplify \\ \\ \frac{1}{x^4} \ / \ negative \ power \ rule \\ \\ Answer: \frac{1}{x^4}
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Use lagrange multipliers to find the point on the plane x â 2y + 3z = 6 that is closest to the point (0, 2, 4).
Arisa [49]
The distance between a point (x,y,z) on the given plane and the point (0, 2, 4) is

\sqrt{f(x,y,z)}=\sqrt{x^2+(y-2)^2+(z-4)^2}

but since \sqrt{f(x,y,z)} and f(x,y,z) share critical points, we can instead consider the problem of optimizing f(x,y,z) subject to x-2y+3z=6.

The Lagrangian is

L(x,y,z,\lambda)=x^2+(y-2)^2+(z-4)^2+\lambda(x-2y+3z-6)

with partial derivatives (set equal to 0)

L_x=2x+\lambda=0\implies x=-\dfrac\lambda2
L_y=2(y-2)-2\lambda=0\implies y=2+\lambda
L_z=2(z-4)+3\lambda=0\implies z=4-\dfrac{3\lambda}2
L_\lambda=x-2y+3z-6=0\implies x-2y+3z=6

Solve for \lambda:

x-2y+3z=-\dfrac\lambda2-2(2+\lambda)+3\left(4-\dfrac{3\lambda}2\right)=6
\implies2=7\lambda\implies\lambda=\dfrac27

which gives the critical point

x=-\dfrac17,y=\dfrac{16}7,z=\dfrac{25}7

We can confirm that this is a minimum by checking the Hessian matrix of f(x,y,z):

\mathbf H(x,y,z)=\begin{bmatrix}f_{xx}&f_{xy}&f_{xz}\\f_{yx}&f_{yy}&f_{yz}\\f_{zx}&f_{zy}&f_{zz}\end{bmatrix}=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}

\mathbf H is positive definite (we see its determinant and the determinants of its leading principal minors are positive), which indicates that there is a minimum at this critical point.

At this point, we get a distance from (0, 2, 4) of

\sqrt{f\left(-\dfrac17,\dfrac{16}7,\dfrac{25}7\right)}=\sqrt{\dfrac27}
8 0
3 years ago
The product of two factors is 7,200.if one of the factors is 90,what is the other factor?
Eddi Din [679]

Ok. Something times 90 = 7200.


Let something = x


90x = 7200


Divide both sides by 90.


90x/90 = 7200/90


x = 80


The other factor is 80.


5 0
3 years ago
Send help 15 and 16 pls
lara [203]

Step-by-step explanation:

15. If f(x) = 20x, adding a $10 fee for cleaning would be

g(x) = 20x + 10

The graph of g(x) is shifted 10 units up from the graph of f(x).

16. f(x) = 2x + 1

g(x) = (2x + 1) + 5

h(x) = 3f(2x + 1)

g(x) shifted f(x) 5 units to the right.

h(x) is 3 times steeper than f(x).

If this is the correct answer, then please give it the brainliest.

5 0
3 years ago
Simplify each expression.
irga5000 [103]

2 \sqrt{ {m}^{7} }  \\ 6 {n}^{2}  \\ 2x \sqrt[3]{2x}
3 0
3 years ago
Read 2 more answers
I need help solving this step by step. Work would be nice so I can look and understand it so I can try a different problem. ​
zimovet [89]

Answer:

I need points, sorry, i cant help

Step-by-step explanation:

3 0
3 years ago
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