Math / Numbers & Operations

QuestionSimplify. $\sqrt{72}$ $\square$ $\sqrt{\square}$
Square root

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of $72$ . Watch out! Don't forget to find the largest perfect square that divides $72$ !

STEP 2

1. Find the prime factorization of $72$ .
2. Simplify the square root.

STEP 3

Let's break down $72$ into its prime factors!
We can start by dividing by $2$ since $72$ is even: $72 = 2 \cdot 36$ .

STEP 4

Now, look at $36$ .
It's also even, so we can divide by $2$ again: $36 = 2 \cdot 18$ .
So far, we have $72 = 2 \cdot 2 \cdot 18$ .

STEP 5

$18$ is also even! $18 = 2 \cdot 9$ .
Now we have $72 = 2 \cdot 2 \cdot 2 \cdot 9$ .

STEP 6

$9$ is $3 \cdot 3$ , so our prime factorization is $72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$ , or $2^3 \cdot 3^2$ .
Awesome!

STEP 7

Now, let's rewrite $\sqrt{72}$ using our prime factorization: $\sqrt{72} = \sqrt{2^3 \cdot 3^2}$ .

STEP 8

Remember, $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ , so we can write $\sqrt{2^3 \cdot 3^2} = \sqrt{2^3} \cdot \sqrt{3^2}$ .

STEP 9

We can rewrite $2^3$ as $2^2 \cdot 2$ , so we have $\sqrt{2^3} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$ .

STEP 10

And $\sqrt{3^2} = 3$ .

STEP 11

Putting it all together, we get $\sqrt{72} = 2\sqrt{2} \cdot 3 = 6\sqrt{2}$ .
We've simplified our square root!

STEP 12

$6\sqrt{2}$

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QuestionSimplify. $\sqrt{72}$ $\square$ $\sqrt{\square}$
Square root

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of $72$ . Watch out! Don't forget to find the largest perfect square that divides $72$ !

STEP 2

1. Find the prime factorization of $72$ .
2. Simplify the square root.

STEP 3

Let's break down $72$ into its prime factors!
We can start by dividing by $2$ since $72$ is even: $72 = 2 \cdot 36$ .

STEP 4

Now, look at $36$ .
It's also even, so we can divide by $2$ again: $36 = 2 \cdot 18$ .
So far, we have $72 = 2 \cdot 2 \cdot 18$ .

STEP 5

$18$ is also even! $18 = 2 \cdot 9$ .
Now we have $72 = 2 \cdot 2 \cdot 2 \cdot 9$ .

STEP 6

$9$ is $3 \cdot 3$ , so our prime factorization is $72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$ , or $2^3 \cdot 3^2$ .
Awesome!

STEP 7

Now, let's rewrite $\sqrt{72}$ using our prime factorization: $\sqrt{72} = \sqrt{2^3 \cdot 3^2}$ .

STEP 8

Remember, $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ , so we can write $\sqrt{2^3 \cdot 3^2} = \sqrt{2^3} \cdot \sqrt{3^2}$ .

STEP 9

We can rewrite $2^3$ as $2^2 \cdot 2$ , so we have $\sqrt{2^3} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$ .

STEP 10

And $\sqrt{3^2} = 3$ .

STEP 11

Putting it all together, we get $\sqrt{72} = 2\sqrt{2} \cdot 3 = 6\sqrt{2}$ .
We've simplified our square root!

STEP 12

$6\sqrt{2}$

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QuestionSimplify. 72\sqrt{72}72​ □\square□ □\sqrt{\square}□​Square root

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of 727272. Watch out! Don't forget to find the *largest* perfect square that divides 727272!

STEP 2

1. Find the prime factorization of 727272.2. Simplify the square root.

STEP 3

Let's **break down** 727272 into its **prime factors**!We can start by dividing by **222** since 727272 is even: 72=2⋅3672 = 2 \cdot 3672=2⋅36.

STEP 4

Now, look at 363636.It's also even, so we can divide by 222 again: 36=2⋅1836 = 2 \cdot 1836=2⋅18.So far, we have 72=2⋅2⋅1872 = 2 \cdot 2 \cdot 1872=2⋅2⋅18.

STEP 5

181818 is also even! 18=2⋅918 = 2 \cdot 918=2⋅9.Now we have 72=2⋅2⋅2⋅972 = 2 \cdot 2 \cdot 2 \cdot 972=2⋅2⋅2⋅9.

STEP 6

999 is 3⋅33 \cdot 33⋅3, so our **prime factorization** is 72=2⋅2⋅2⋅3⋅372 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 372=2⋅2⋅2⋅3⋅3, or 23⋅322^3 \cdot 3^223⋅32.Awesome!

STEP 7

Now, let's **rewrite** 72\sqrt{72}72​ using our prime factorization: 72=23⋅32\sqrt{72} = \sqrt{2^3 \cdot 3^2}72​=23⋅32​.

STEP 8

Remember, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}a⋅b​=a​⋅b​, so we can write 23⋅32=23⋅32\sqrt{2^3 \cdot 3^2} = \sqrt{2^3} \cdot \sqrt{3^2}23⋅32​=23​⋅32​.

STEP 9

We can **rewrite** 232^323 as 22⋅22^2 \cdot 222⋅2, so we have 23=22⋅2=22⋅2=22\sqrt{2^3} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}23​=22⋅2​=22​⋅2​=22​.

STEP 10

And 32=3\sqrt{3^2} = 332​=3.

STEP 11

Putting it all together, we get 72=22⋅3=62\sqrt{72} = 2\sqrt{2} \cdot 3 = 6\sqrt{2}72​=22​⋅3=62​.We've **simplified** our square root!

STEP 12

626\sqrt{2}62​

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QuestionSimplify. $\sqrt{72}$ $\square$ $\sqrt{\square}$
Square root

What is this asking? We need to simplify the square root of $72$ . Watch out! Don't forget to find the largest perfect square that divides $72$ !

1. Find the prime factorization of $72$ .
2. Simplify the square root.

Let's break down $72$ into its prime factors!
We can start by dividing by $2$ since $72$ is even: $72 = 2 \cdot 36$ .

Now, look at $36$ .
It's also even, so we can divide by $2$ again: $36 = 2 \cdot 18$ .
So far, we have $72 = 2 \cdot 2 \cdot 18$ .

$18$ is also even! $18 = 2 \cdot 9$ .
Now we have $72 = 2 \cdot 2 \cdot 2 \cdot 9$ .

$9$ is $3 \cdot 3$ , so our prime factorization is $72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$ , or $2^3 \cdot 3^2$ .
Awesome!

Now, let's rewrite $\sqrt{72}$ using our prime factorization: $\sqrt{72} = \sqrt{2^3 \cdot 3^2}$ .

Remember, $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ , so we can write $\sqrt{2^3 \cdot 3^2} = \sqrt{2^3} \cdot \sqrt{3^2}$ .

We can rewrite $2^3$ as $2^2 \cdot 2$ , so we have $\sqrt{2^3} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$ .

And $\sqrt{3^2} = 3$ .

Putting it all together, we get $\sqrt{72} = 2\sqrt{2} \cdot 3 = 6\sqrt{2}$ .
We've simplified our square root!

$6\sqrt{2}$