Question

Simplify each radical And show your work √72 √20 √42 2√12 3√75 -2√18​

Answer 1

Answer:

[See Below]

Step-by-step explanation:

$\boxed{Hey~There!}$

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$\boxed{Simplify~each~radical:}$

1.) $\sqrt{72}$:

• Factor $72$ into its prime factors :

          $72 = 23 * 32$  

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• Factors which will be extracted are:

          $36 = 22 * 32$  

• Factors which will remain inside the root are:

          $2 = 2$  

• To complete the simplification we take the square root of the factors which are to be extracted. We do this by dividing their exponents by $2$:

          $6 = 2 * 3$

$\boxed{The~answer~is: 6\sqrt{2}}$

2.) $\sqrt{20}$:

• Factor $20$ into its prime factors :

          $20 = 22 * 5$  

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• Factors which will be extracted are:

         $4 = 22$

• Factors which will remain inside the root are:

          $5 = 5$  

• To complete the simplification we take the square root of the factors which are to be extracted. We do this by dividing their exponents by $2$:

          $2 = 2$

$\boxed{The~answer~is: 2\sqrt{5}}$

3.) $\sqrt{42}$:

• Factor $42$ into its prime factors :

          $42 = 2 * 3 * 7$

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• In our case however, all the factors are only raised to the first power and this means that the square root can not be simplified .

$\boxed{The~answer~is: \sqrt{42} }$

4.) $2\sqrt{12}$:

• Factor $12$ into its prime factors :

          $12 = 22 * 3$  

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• Factors which will be extracted are:

          $4 = 22$  

• Factors which will remain inside the root are:

          $3 = 3$  

• To complete the simplification we take the square root of the factors which are to be extracted. We do this by dividing their exponents by $2$:

          $2 = 2$

$\boxed{The~answer~is: 4\sqrt{3} }$

5.) $3\sqrt{75}$:

• Factor $75$ into its prime factors :

          $75 = 3 * 52$  

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• Factors which will be extracted are:

          $25 = 52$  

• Factors which will remain inside the root are:

          $3 = 3$  

• To complete the simplification we take the square root of the factors which are to be extracted. We do this by dividing their exponents by $2$:

          $5 = 5$

$\boxed{The~answer~is: 15\sqrt{3}}$

6.) $-2\sqrt{18}$:

• Factor $18$ into its prime factors :

          $18 = 2 * 32$  

• To simplify a square root, we extract factors which are squares, factors that are raised to an even exponent.

• Factors which will be extracted are:

         $9 = 32$  

• Factors which will remain inside the root are:

          $2 = 2$  

• To complete the simplification we take the square root of the factors which are to be extracted. We do this by dividing their exponents by $2$:

          $3 = 3$

$\boxed{The~answer~is: 3\sqrt{5}~or~-6\sqrt{2}}$

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$Hope~this~helps~Mate~\\-Your~Friendly~Answerer,~Shane$  ^-^