Let assume that $\sqrt7$ is a rational number. Therefore it can be expressed as a fraction $\dfrac{a}{b}$ where $a,b\in\mathbb{Z}$ and $\text{gcd}(a,b)=1$ .

$\sqrt7=\dfrac{a}{b}\\\\7=\dfrac{a^2}{b^2}\\\\a^2=7b^2$

This means that $a^2$ is divisible by 7, and therefore also $a$ is divisible by 7.

So, $a=7k$ where $k\in\mathbb{Z}$

$(7k)^2=7b^2\\\\49k^2=7b^2\\\\7k^2=b^2$

Analogically to $a^2=7b^2$ ------- $b^2$ is divisible by 7 and therefore so is $b$ .

But if both numbers $a$ and $b$ are divisible by 7, then $\text{gcd}(a,b)=7$ which contradicts our earlier assumption that $\text{gcd}(a,b)=1$ .

Therefore $\sqrt7$ is an irrational number.

8 0

4 years ago

Which of the following is not equal to sec⁡(-85°)?

MAVERICK [17]

-85 degrees in is the 4th quadrant so the cos and sec will be positive
sec is also positive in first quadrant so sec(85) will be equal to sec(-85)
Sec is is negative in 3rd quadrant so -sec(-95) will be same also.

sec 275 is in 3rd quadrant so will be negative so not equal
also sec(-95) will be same angle as sec 275

the last 2 options are not equal to sec(-85)

7 0

3 years ago

Read 2 more answers

A taxi service charges a flat fee of $1.25 and $0.75 per mile. If Henri has $14.00, which of the following shows the number of m

Vanyuwa [196]

Answer:

Option A M<17 m is less than or equal to 17

Step-by-step explanation:

4 0

3 years ago

Lorena is making a gelatin dessert. The recipe calls for

slavikrds [6]

Answer:

4 2/3 cups

Step-by-step explanation:

Add 2+2 and 1/3+1/3

5 0

4 years ago