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3 years ago

Simplify the expression

Mathematics

2 answers:

goblinko [34]3 years ago

8 0

Answer:

12 to the second power.

Step-by-step explanation:

Liono4ka [1.6K]3 years ago

6 0

Answer:

Your answer is 144

Step-by-step explanation:

11-3=8

8+4=12

12 to the second power is 144

Hope this helps

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How to solve part ii and iii

iragen [17]

(i) Given that

$\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(xy) = \dfrac{7\pi}{12}$

when $x=1$ this reduces to

$\tan^{-1}(1) + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}$

$\dfrac\pi4 + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}$

$2 \tan^{-1}(y) = \dfrac\pi3$

$\tan^{-1}(y) = \dfrac\pi6$

$\tan\left(\tan^{-1}(y)\right) = \tan\left(\dfrac\pi6\right)$

$\implies \boxed{y = \dfrac1{\sqrt3}}$

(ii) Differentiate $\tan^{-1}(xy)$ implicitly with respect to $x$ . By the chain and product rules,

$\dfrac d{dx} \tan^{-1}(xy) = \dfrac1{1+(xy)^2} \times \dfrac d{dx}xy = \boxed{\dfrac{y + x\frac{dy}{dx}}{1 + x^2y^2}}$

(iii) Differentiating both sides of the given equation leads to

$\dfrac1{1+x^2} + \dfrac1{1+y^2} \dfrac{dy}{dx} + \dfrac{y + x\frac{dy}{dx}}{1+x^2y^2} = 0$

where we use the result from (ii) for the derivative of $\tan^{-1}(xy)$ .

Solve for $\frac{dy}{dx}$ :

$\dfrac1{1+x^2} + \left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} + \dfrac y{1+x^2y^2} = 0$

$\left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} = -\left(\dfrac1{1+x^2} + \dfrac y{1+x^2y^2}\right)$

$\dfrac{1+x^2y^2 + x(1+y^2)}{(1+y^2)(1+x^2y^2)} \dfrac{dy}{dx} = - \dfrac{1+x^2y^2 + y(1+x^2)}{(1+x^2)(1+x^2y^2)}$

$\implies \dfrac{dy}{dx} = - \dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2) (1 + x^2y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2) (1+x^2y^2)}$

$\implies \dfrac{dy}{dx} = -\dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2)}$

From part (i), we have $x=1$ and $y=\frac1{\sqrt3}$ , and substituting these leads to

$\dfrac{dy}{dx} = -\dfrac{\left(1 + \frac13 + \frac1{\sqrt3} + \frac1{\sqrt3}\right) \left(1 + \frac13\right)}{\left(1 + \frac13 + 1 + \frac13\right) \left(1 + 1\right)}$

$\dfrac{dy}{dx} = -\dfrac{\left(\frac43 + \frac2{\sqrt3}\right) \times \frac43}{\frac83 \times 2}$

$\dfrac{dy}{dx} = -\dfrac13 - \dfrac1{2\sqrt3}$

as required.

3 0

2 years ago

Determine the transformations done on pentagon 1 to obtain pentagon 2.<br> [[ ANSWER ASAP ]]

topjm [15]

Answer:first option

Step-by-step explanation:

Is enlarged by scale factor of 2 and is moved to another place

4 0

3 years ago

5y=-3+15<br><br> convert to slope intercept form

creativ13 [48]

Answer:

y=-⅗x+3

Step-by-step explanation:

5y=-3x+15. Divide 5 on both sides

5. 5

y=-⅗x+15/5

y=-⅗x+3

Hope this helps :)

8 0

3 years ago

Solve (-8) to the second power

labwork [276]

Answer:

Step-by-step explanation:

8 times 8 is 64, but it is the opposite of positive is negative

3 0

3 years ago

Read 2 more answers

For each of the following pairs of signed integers (2’s complement), subtract the second from the first. Show the decimal equiva

Dmitry_Shevchenko [17]

Answer:

The answers are

a)111010 - 001111 = 101011 = 43

b) 000100 - 011000 = 111111111 (overflow) = -20

c) 010001 - 011010 = 11111111 (overflow) = -9

d) 010000 - 100100 = 1111111 (overflow) = -20

Step-by-step explanation:

a)111010 = 1×2⁵+1×2⁴+1×2³+0×2²+1×2¹+0×2⁰ = 32+16+8+0+2+0= 58

001111 = 0×2⁵+0×2⁴+1×2³+1×2²+1×2¹+1×2⁰ = 8+4+2+1=15

111010 - 001111 = 101011 = 43

b) 000100 = 0×2⁵+0×2⁴+0×2³+1×2²+0×2¹+0×2⁰ = 4

011000 = 0×2⁵+1×2⁴+1×2³+0×2²+0×2¹+0×2⁰ = 24

000100 - 011000 = 1111111111 (overflow), 4 - 24 =-20

c) 010001 = 1×2⁴+1×2⁰ = 17

011010 = 1×2⁴+1×2³+1×2¹ = 26

010001 - 011010 = 11111111 (overflow) 17-26=-9

d) 010000 = 1×2⁴ = 16

100100 = 1×2⁵+1×2² = 36

010000 - 100100 = 1111111 (overflow) 16-36=-20

3 0

3 years ago