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3 years ago

23−(−7) Please help me

Mathematics

1 answer:

adell [148]3 years ago

6 0

30 positive

When there is a subtraction sign with a negative number it becomes an addition sign

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The value of (a+b)² - (a-b)² is

ANEK [815]

Answer:

4ab

Step-by-step explanation:

(a+b)^2-(a-b)^2

=a^2 + 2ab + b^2 -(a^2 - 2ab +b^2)

=a^2 + 2ab + b^2 - a^2 + 2ab - b^2

=2ab + 2ab

=4ab

5 0

3 years ago

Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64

tino4ka555 [31]

$x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}$

By Fermat's little theorem, we have

$x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5$

$x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7$

5 and 7 are both prime, so $\varphi(5)=4$ and $\varphi(7)=6$ . By Euler's theorem, we get

$x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5$

$x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7$

Now we can use the Chinese remainder theorem to solve for $x$ . Start with

$x=2\cdot7+5\cdot6$

Taken mod 5, the second term vanishes and $14\equiv4\pmod5$ . Multiply by the inverse of 4 mod 5 (4), then by 2.

$x=2\cdot7\cdot4\cdot2+5\cdot6$

Taken mod 7, the first term vanishes and $30\equiv2\pmod7$ . Multiply by the inverse of 2 mod 7 (4), then by 6.

$x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6$

$\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}$

$x^5\equiv3\pmod{64}$

We have $\varphi(64)=32$ , so by Euler's theorem,

$x^{32}\equiv1\pmod{64}$

Now, raising both sides of the original congruence to the power of 6 gives

$x^{30}\equiv3^6\equiv729\equiv25\pmod{64}$

Then multiplying both sides by $x^2$ gives

$x^{32}\equiv25x^2\equiv1\pmod{64}$

so that $x^2$ is the inverse of 25 mod 64. To find this inverse, solve for $y$ in $25y\equiv1\pmod{64}$ . Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that $(-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}$ .

So we know

$25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}$

Squaring both sides of this gives

$x^4\equiv1681\equiv17\pmod{64}$

and multiplying both sides by $x$ tells us

$x^5\equiv17x\equiv3\pmod{64}$

Use the Euclidean algorithm to solve for $x$ .

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that $(-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}$ , and so $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

5 0

3 years ago

What is the slope of the line that passes through the points (12,-8) and (-9,-11)

Sauron [17]

Answer: 1/7

Explanation:

1. Use the formula, y2-y1/x2-x1
Plug in the points, (12, -8) and (-9, -11) into the formula.
Point (12, -8): 12 is the X1 and -8 is the y1.
Point (-9, -11): -9 is the X2 and -11 is the y2
So, -11--8/-9-12

2. -11--8/-9-12= -3/-21= 1/7

I hope this helps!!

3 0

3 years ago

I found this answer on the Internet. *If* it were an original answer, below is how I would grade it and why.

Dennis_Churaev [7]

Answer:

1) It is given that line AB is tangent to the circle at A.

∴ ∠CAB = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

Thus, the measure of ∠CAB is 90º.

4 0

3 years ago

The difference of the product of 3 and m<br> minus the quotient of m divided by 2<br> m=6

s2008m [1.1K]

Answer:5m/2

Step-by-step explanation:

3m-m/2

(2×3m-1×m)/2

(6m-m)/2=5m/2

5 0

3 years ago