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

3 years ago

Today's temperature is 3 degrees warmer than yersterday's temperature t . write an expression for today's temperature

vredina [299]

The formula would be t + 3.

7 0

3 years ago

Given f ( x ) = x 2 , after performing the following transformations: shift upward 76 units and shift 63 units to the right, the

Luden [163]

Answer:

f(x)= (x^2 - 63) + 76

Step-by-step explanation:

4 0

2 years ago

How many integers are there…

Sliva [168]

Answer:

there are infinite integers.

3 0

2 years ago