What is 2×(3×3)÷3-5 and why

Answers

Answer 1

Answer: 1 because if you use order of operations you would:
parenthesis
then multiplication
then division
the subtraction

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Estimate the sum of 196 and 482

Answers

196
+
482
______
678 (HA!)

The phrases "sum of" & "and" indicates this is an addition problem

196 -> 200

482 -> 480 or 500

200+480=680

100+500=600

Why is the product of a rational number and an irrational number irrational

Answers

Great question. Let's let r be a rational number and s be irrational. Note r has to be nonzero for this to work. In other words, it's not true that when we multiply zero, a rational number, by an irrational number like π we get an irrational number. We of course get zero.

The question is: why is the product

$p = rs$

irrational?

In math "why" questions are usually answered with an illuminating proof. Here the indirect proof is enlightening.

Suppose p was rational. Then

$s = \frac p r$

would be rational as well, being the ratio of two rational numbers, so ultimately the ratio of two integers.

But we're given that s is irrational so we have our contradiction and must conclude our assumption that p is rational is false, that is, we conclude p is irrational.

A proof by contradiction.

Let assume that the product of a rational number and an irrational number is rational.

Let $(a)/(b)$ and $(c)/(d)$ be rational numbers, where $a,b,c,d\in \mathbb{Z} \wedge b,d\not=0$ and $x$ an irrational number.

Then

$(a)/(b)\cdot x=(c)/(d)\nx=(bc)/(ad)$

Integers are closed under multiplication, therefore $bc$ and $ad$ are integers, making the number $x=(bc)/(ad)$ rational, which is contradictory with the earlier statement that $x$ is an irrational number.