Question

Use a factor tree to find the prime factorization of the given number. Use exponents in your answer when appropriate. 84

Short answer

The prime factorization of 84 is \[ 2^2 \times 3 \times 7 \].

Step-by-step solution

Start with the given number

Begin with the number 84 and find two factors of 84 that when multiplied give the original number. For example, 84 can be factored into 2 and 42.

Factor the result further

Take the larger factor from Step 1, which is 42, and find its factors. 42 can be factored into 2 and 21.

Continue factoring

Now, take the factor 21 and break it down into 3 and 7. Both 3 and 7 are prime numbers.

Collect all prime factors

Now, all numbers are prime: 2, 2, 3, and 7. These are the prime factors of 84.

Write the prime factorization using exponents

Group the prime factors and use exponents where appropriate. The prime factorization of 84 is written as \[ 2^2 \times 3 \times 7 \].

Key concepts

Factor Tree

A factor tree is a simple way to break down a number into its prime factors. Imagine a tree with branches that double with each factor you find.
Start with the main number, like 84, and split it into two factors. For example, 84 becomes 2 and 42. Keep breaking down each non-prime factor into smaller factors until all you have left are prime numbers.
When creating a factor tree, visualize it like a family tree, where each branch leads to two smaller branches until you can't split them anymore. It helps organize your work and ensures you miss no steps.

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number is greater than 1 and has no divisors other than 1 and itself.
Examples include 2, 3, 5, 7, 11, and so on. When you're factoring a number, knowing your primes helps you know when to stop breaking down factors.
In the case of 84, your factor tree stops when you hit 2, 3, and 7—all prime numbers. Always check if a number you're factoring is prime; if it is, you can stop there.

Exponents

Exponents are a shorthand way to show repeated multiplication. If you have the number 2 appearing twice in your list of prime factors, instead of writing it as 2, 2, you write it as \( 2^2 \).
For example, in the prime factorization of 84, you have two 2s, one 3, and one 7. So, it's simplified as: \[ 2^2 \space \times \space 3 \space \times \space 7 \].
Exponents make the expression cleaner and easier to read, especially for larger numbers with many repeated prime factors.