Math  /  Numbers & Operations

Question4. Draw a factor tree. 144144

Studdy Solution

STEP 1

What is this asking? We need to break down the number 144 into its smallest prime factors using a factor tree. Watch out! Make sure every branch ends in a prime number!
Don't stop early!

STEP 2

1. Find a Factor Pair
2. Factor the First Branch
3. Factor the Second Branch
4. Gather the Primes

STEP 3

Let's **start** with our number: 144144.
We need to find two numbers that multiply together to give us 144144.
How about 1212 and 1212? 1212=14412 \cdot 12 = 144.
Perfect! We'll draw two branches coming down from 144144, one with 1212 and the other with 1212.

STEP 4

Now, let's look at the first 1212.
What two numbers multiply to 1212?
Let's use 33 and 44.
So, two branches come down from the 1212, one with 33 and one with 44.
Since 33 is a prime number, that branch is done!

STEP 5

What about 44? 44 can be broken down into 22 and 22. 22 is also a prime number, so both of those branches are finished!

STEP 6

We have another 1212 to break down!
We can do the same thing as before. 1212 is 33 times 44.
So, two branches come down from the 1212, one with 33 and one with 44. 33 is prime, so that branch is done!

STEP 7

And just like before, the 44 becomes 22 and 22, both of which are prime!
All branches have reached prime numbers!

STEP 8

Now, let's gather all the prime numbers at the ends of the branches.
We have two 33s and four 22s.
So, 144144 can be written as 2222332 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3.

STEP 9

We can write this using exponents as 24322^4 \cdot 3^2. 22 multiplied by itself four times (242^4) is 2222=162 \cdot 2 \cdot 2 \cdot 2 = 16. 33 multiplied by itself twice (323^2) is 33=93 \cdot 3 = 9.
And 169=14416 \cdot 9 = 144!

STEP 10

The prime factorization of 144144 is 24322^4 \cdot 3^2.
The factor tree shows all the prime factors and how they multiply together to make 144144.

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