Math / Numbers & Operations

Question $\sqrt{1620}$

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of 1620. Watch out! Don't forget to look for perfect squares hiding inside!

STEP 2

1. Find the prime factorization.
2. Simplify the radical.

STEP 3

Let's break down $\sqrt{1620}$ by finding its prime factorization!
This is like finding the secret ingredients of our number.
We're looking for the smallest prime numbers multiplied together to make 1620.

STEP 4

We can start by dividing 1620 by 2, which gives us 810.
So, $1620 = 2 \cdot 810$ .

STEP 5

810 is also divisible by 2, giving us 405.
Now we have $1620 = 2 \cdot 2 \cdot 405$ .

STEP 6

405 is not divisible by 2, but it is divisible by 3. $405 = 3 \cdot 135$ , so $1620 = 2 \cdot 2 \cdot 3 \cdot 135$ .

STEP 7

135 is also divisible by 3, giving us 45.
So, $1620 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 45$ .

STEP 8

And guess what?
45 is divisible by 3 again! $45 = 3 \cdot 15$ , which means $1620 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 15$ .

STEP 9

Finally, $15 = 3 \cdot 5$ .
So, the prime factorization of 1620 is $2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5$ , or $2^2 \cdot 3^4 \cdot 5$ .
Awesome!

STEP 10

Now, let's rewrite our radical using the prime factorization: $\sqrt{1620} = \sqrt{2^2 \cdot 3^4 \cdot 5}$ .

STEP 11

Remember, the square root of a product is the product of the square roots.
So, we can rewrite this as $\sqrt{2^2} \cdot \sqrt{3^4} \cdot \sqrt{5}$ .

STEP 12

The square root of $2^2$ is 2.
The square root of $3^4$ is $3^2$ , which is 9.
And $\sqrt{5}$ stays as it is because 5 is a prime number.

STEP 13

Putting it all together, we get $2 \cdot 9 \cdot \sqrt{5}$ , which simplifies to $18\sqrt{5}$ .
Boom!

STEP 14

The simplified form of $\sqrt{1620}$ is $18\sqrt{5}$ .

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