Math / Numbers & Operations

QuestionSimplify. $\sqrt{24}$

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of $24$ .
Basically, we want to find the simplest way to write this radical. Watch out! Don't just grab your calculator!
We want the exact simplified radical form, not a rounded decimal.

STEP 2

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

STEP 3

Let's break down $\sqrt{24}$ by finding the prime factorization of $24$ .
This means we want to write $24$ as a product of its prime factors.
Remember, a prime number is a whole number greater than $1$ that has only two divisors: $1$ and itself.

STEP 4

We can start by dividing $24$ by the smallest prime number, $2$ .
We get $24 = 2 \cdot 12$ .

STEP 5

$12$ isn't prime, so we keep going! $12 = 2 \cdot 6$ , so now we have $24 = 2 \cdot 2 \cdot 6$ .

STEP 6

$6$ is still not prime! $6 = 2 \cdot 3$ .
Finally, we have $24 = 2 \cdot 2 \cdot 2 \cdot 3$ , and all of these factors are prime!
We can write this more compactly using exponents: $24 = 2^3 \cdot 3$ .

STEP 7

Now, let's rewrite our radical using the prime factorization: $\sqrt{24} = \sqrt{2^3 \cdot 3}$ .

STEP 8

Remember that the square root of a product is the product of the square roots: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ .
So, we can write $\sqrt{2^3 \cdot 3} = \sqrt{2^3} \cdot \sqrt{3}$ .

STEP 9

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

STEP 10

Using the rule from before, $\sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2}$ .

STEP 11

Since $\sqrt{2^2} = 2$ , we have $\sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$ .

STEP 12

Putting it all together, we get $\sqrt{24} = \sqrt{2^3 \cdot 3} = \sqrt{2^3} \cdot \sqrt{3} = 2\sqrt{2} \cdot \sqrt{3} = 2\sqrt{2 \cdot 3} = 2\sqrt{6}$ .

STEP 13

Our simplified radical is $2\sqrt{6}$ !

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms