Simplify the following expression 11^7/11^4

a) Assume that n is not odd. Then n is even, that is, n=2k for some integer k. Hence n²=4k²=2(2k²)=2t for some integer t=2k². Then n² is even, therefore n² is not odd. We have proved the counterpositive of this statement.

b) Assume that n is not even, then n is odd. Thus, n=2k+1 for some integer k. Now, n³=(2k+1)³=8k³+6k²+6k+1=2(4k³+3k²+3k)+1=2t+1 for the integer t=4k³+3k²+3k. Thus n³ is odd, that is, n³ is not even.

c) Suppose that n is not odd, that is, n is even. Now, n=2k for some integer k. Then 5n+3=10k+3=2(5k+1)+1, thus 5n+3 is odd, then 5n+3 is not even.

d) Suppose that n is not odd, then n is even. Now, n=2k for some integer k. Then n²-2n+7=4k²-4k+7=2(2k²-2k+3)+1. Hence n²-2n+7 is odd, that is, n²-2n+7 is not even.

e) Assume that -r is not irrational, then -r is rational. Since -1 is rational, then (-1)(-r)=r is rational. Thus r is not irrational.

f) Assume that 1/z is not irrational. Then 1/z is rational. Multiplucative inverses of rational numbers are rational, hence z is rational, that is, z is not irrational.

g) Suppose that z>y. We will prove that z³+zy²≤z²y+y³ is false, that is, we will prove that z³+zy²>z²y+y³. Multiply by the nonnegative number z² in the inequality z>y to get z³>z²y (here we assume z and y nonzero, in this case either z³>0=y³ is true or z³=0>y³ is true). On the other hand, multiply by z² (positive number) to get zy²>y³. Add both inequalities to obtain z³+zy²>z²y+y³ as required.

h) Suppose than n is even. Then n=2k, and n²=4k² is divisible by 4.

i) Assume that "z is irrational or y is irrational" is false. Then z is rational and y is rational. Rational numbers are closed under sum, then z+y is rational, that is, z+y is not irrational.

3 0

3 years ago

I need help with the promblem.

Viefleur [7K]

Six dozen cupcakes is 72. So in this case Kailyn made 88 cupcakes, but that was 72 more than Melissa made so Melissa only made 16.

3 0

3 years ago

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The speed of a ship is given by d3, where d is the distance the ship travels in 3 hours. if the ship travels 48 miles in 3 hours

Kay [80]

Hello there! Thank you for asking your question here at Brainly! I will be assisting you today with answering this problem, and will teach you how to handle it on your own in the future.

First, let's take a look at our problem.
"The speed of a ship is given by D3, where D is the distance the ship travels in 3 hours. If the ship travels 48 miles in 3 hours, what is the speed of the ship?"

To solve for this, we need to understand the formula for speed.
The formula for speed is:
Speed = $\frac{Distance}{Time}$

Now that we have our formula, let's define our variables.

So, we have our distance, 48 miles, and our time, 3 hours.
Let's plug in these numbers to the formula.
Speed = $\frac{48}{3}$

Simplify the fraction to get the speed.
$\frac{48}{3}$ = 16.

Your final equation is:
Speed = 16.

Our final answer is 16 mph.

I hope this helps!

4 0

3 years ago

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