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tiny-mole [99]
3 years ago
15

Marsha used 6 cups of flour to make 4 trays of

Mathematics
1 answer:
PolarNik [594]3 years ago
6 0

Answer:

9

Step-by-step explanation:

4/6 = 1.5, 1.5(6)= 9

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Which of the following functions are homomorphisms?
Vikentia [17]
Part A:

Given f:Z \rightarrow Z, defined by f(x)=-x

f(x+y)=-(x+y)=-x-y \\  \\ f(x)+f(y)=-x+(-y)=-x-y

but

f(xy)=-xy \\  \\ f(x)\cdot f(y)=-x\cdot-y=xy

Since, f(xy) ≠ f(x)f(y)

Therefore, the function is not a homomorphism.



Part B:

Given f:Z_2 \rightarrow Z_2, defined by f(x)=-x

Note that in Z_2, -1 = 1 and f(0) = 0 and f(1) = -1 = 1, so we can also use the formular f(x)=x

f(x+y)=x+y \\  \\ f(x)+f(y)=x+y

and

f(xy)=xy \\  \\ f(x)\cdot f(y)=xy

Therefore, the function is a homomorphism.



Part C:

Given g:Q\rightarrow Q, defined by g(x)= \frac{1}{x^2+1}

g(x+y)= \frac{1}{(x+y)^2+1} = \frac{1}{x^2+2xy+y^2+1}  \\  \\ g(x)+g(y)= \frac{1}{x^2+1} + \frac{1}{y^2+1} = \frac{y^2+1+x^2+1}{(x^2+1)(y^2+1)} = \frac{x^2+y^2+2}{x^2y^2+x^2+y^2+1}

Since, f(x+y) ≠ f(x) + f(y), therefore, the function is not a homomorphism.



Part D:

Given h:R\rightarrow M(R), defined by h(a)=  \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)

h(a+b)= \left(\begin{array}{cc}-(a+b)&0\\a+b&0\end{array}\right)= \left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right) \\  \\ h(a)+h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)+ \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)=\left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right)

but

h(ab)= \left(\begin{array}{cc}-ab&0\\ab&0\end{array}\right) \\  \\ h(a)\cdot h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)\cdot \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)= \left(\begin{array}{cc}ab&0\\-ab&0\end{array}\right)

Since, h(ab) ≠ h(a)h(b), therefore, the funtion is not a homomorphism.



Part E:

Given f:Z_{12}\rightarrow Z_4, defined by \left([x_{12}]\right)=[x_4], where [u_n] denotes the lass of the integer u in Z_n.

Then, for any [a_{12}],[b_{12}]\in Z_{12}, we have

f\left([a_{12}]+[b_{12}]\right)=f\left([a+b]_{12}\right) \\  \\ =[a+b]_4=[a]_4+[b]_4=f\left([a]_{12}\right)+f\left([b]_{12}\right)

and

f\left([a_{12}][b_{12}]\right)=f\left([ab]_{12}\right) \\ \\ =[ab]_4=[a]_4[b]_4=f\left([a]_{12}\right)f\left([b]_{12}\right)

Therefore, the function is a homomorphism.
7 0
3 years ago
Solve for x: 3√5x² +25x-10√√5 = 0
bogdanovich [222]

Answer:  x = - 2√5, √5/3

Step-by-step explantion

3 √ 5 x 2 + 25 x − 10 √ 5 = 0 35x2+25x−105=0

⇒ 3√5x2 + 30x – 5x – 10√5 = 0  

⇒ 3√5x(x + 2√5) – 5(x + 2√5) = 0  

⇒ (x + 2√5)(3√5x – 5) = 0

x = - 2√5, √5/3

3 0
1 year ago
Read 2 more answers
Write each of the following as a polynomial in descending powers of its variable
nikklg [1K]
1. 7x(x + 3)
    7x(x) + 7(3)
    7x² + 21

2. (7x + 1)(x + 3)
    7x² + 21x + x + 3
    7x² + 22x + 3

3. (x² + x)²
    (x² + x)(x² + x)
   x^4 + x³ + x³ + x²
   x^4 + 2x³ + x²
4 0
3 years ago
What is 8 divided by 108
patriot [66]

The answer is 13.5, hope this helps!

6 0
3 years ago
Read 2 more answers
If r = 4 and ^a8 = 100, what is the first term of the geometric<br> sequence?
telo118 [61]

The first term of the geometric  sequence is 0.0061

<u>Step-by-step explanation:</u>

The general form of geometric  sequence is a, ar, ar²,ar³,.......

where,

  • a is the first term of the sequence.
  • r is the common ratio. Here, the common ratio r = 4.
  • The 8th term of the sequence is 100. Hence n = 8.

<u>The formula to find the nth term of the geometric sequence is given by :</u>

⇒ n_{th} term = ar^{n-1}

⇒ 8_{th}term  = a\times4^{8-1}

⇒ 100 = a \times 4^{7}

⇒ 100 = a \times 16384

⇒ a = \frac{100}{16384}

⇒ a = 0.0061

Therefore, the first term of the geometric  sequence is 0.0061

6 0
3 years ago
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