Question

when simplifying a radical, and you get an answer like \\(\sqrt{49}\\), you usually simplify it to 7. you can also simplify radical expressions that are not a perfect square, like \\(\sqrt{50}\\). we consider an expression simplified when the value under the square root cannot be divided by a perfect square. here’s how you simplify a radical expression.
to simplify \\(\sqrt{50}\\), look at all the perfect squares under 50, and find the largest one that will go evenly into 50. 49? no. 36? no. 25? yes!
now you try!

1. \\(\sqrt{12}\\) \\(\implies\\) \\(\sqrt{\square * \square}\\) \\(\implies\\) \\(\sqrt{\square} * \sqrt{\square}\\) \\(\implies\\) \\(\square\sqrt{\square}\\)

\\(\quad\quad\quad\quad\quad\\) the largest perfect square that goes into 12. \\(\quad\quad\quad\quad\quad\\) simplify the perfect square part. the rest stays under the square root sign.

1. \\(\sqrt{18}\\) \\(\implies\\) \\(\sqrt{\square * \square}\\) \\(\implies\\) \\(\sqrt{\square} * \sqrt{\square}\\) \\(\implies\\) \\(\square\sqrt{\square}\\)

\\(\quad\quad\quad\quad\quad\\) the largest perfect square that goes into 18. \\(\quad\quad\quad\quad\quad\\) simplify the perfect square part. the rest stays under the square root sign.

1. \\(\sqrt{24}\\) \\(\implies\\) \\(\sqrt{\square * \square}\\) \\(\implies\\) \\(\sqrt{\square} * \sqrt{\square}\\) \\(\implies\\) \\(\square\sqrt{\square}\\)
2. \\(\sqrt{75}\\) \\(\implies\\) \\(\sqrt{\square * \square}\\) \\(\implies\\) \\(\sqrt{\square} * \sqrt{\square}\\) \\(\implies\\) \\(\square\sqrt{\square}\\)

Explanation:

Step1: Factor out largest perfect square

For $\sqrt{12}$: $\sqrt{4*3}$

Step2: Split into product of roots

$\sqrt{4}*\sqrt{3}$

Step3: Simplify perfect square

$2\sqrt{3}$

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Step1: Factor out largest perfect square

For $\sqrt{18}$: $\sqrt{9*2}$

Step2: Split into product of roots

$\sqrt{9}*\sqrt{2}$

Step3: Simplify perfect square

$3\sqrt{2}$

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Step1: Factor out largest perfect square

For $\sqrt{24}$: $\sqrt{4*6}$

Step2: Split into product of roots

$\sqrt{4}*\sqrt{6}$

Step3: Simplify perfect square

$2\sqrt{6}$

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Step1: Factor out largest perfect square

For $\sqrt{75}$: $\sqrt{25*3}$

Step2: Split into product of roots

$\sqrt{25}*\sqrt{3}$

Step3: Simplify perfect square

$5\sqrt{3}$

Answer:

1. $\sqrt{4*3}$, $\sqrt{4}*\sqrt{3}$, $2\sqrt{3}$
2. $\sqrt{9*2}$, $\sqrt{9}*\sqrt{2}$, $3\sqrt{2}$
3. $\sqrt{4*6}$, $\sqrt{4}*\sqrt{6}$, $2\sqrt{6}$
4. $\sqrt{25*3}$, $\sqrt{25}*\sqrt{3}$, $5\sqrt{3}$