Is the product of $3 \sqrt{2} and 8 \sqrt{18}$ $3\sqrt 2 and 8\sqrt 18$ rational or irrational?

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Answer 1 · 2017-11-13T16:05:35

See a solution process below:

First, rewrite the expression as:
$3 \sqrt{2} \times 8 \sqrt{18} \Rightarrow$ $3sqrt(2) xx 8sqrt(18) =>$

$(3 \times 8) \sqrt{2} \sqrt{18} \Rightarrow$ $(3 xx 8)sqrt(2)sqrt(18) =>$

$24 \sqrt{2} \sqrt{18}$ $24sqrt(2)sqrt(18)$

Next, we can use this rule for radicals to combine the radicals in the expression:

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ $sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))$

$24 \sqrt{2} \cdot \sqrt{18} \Rightarrow$ $24sqrt(color(red)(2)) * sqrt(color(blue)(18)) =>$

$24 \sqrt{2 \cdot 18} \Rightarrow$ $24sqrt(color(red)(2) * color(blue)(18)) =>$

$24 \sqrt{36} \Rightarrow$ $24sqrt(36) =>$

$24 \cdot 6 \Rightarrow$ $24 * 6 =>$

$144$ $144$

A rational number can be written as a fraction of 2 integers:

$144 = \frac{144}{1}$ $144 = 144/1$

Therefore the expression in the problem is rational.

Is the product of 3√2and8√183\sqrt 2 and 8\sqrt 18 rational or irrational?