# How to Solve Prime Factorization with Exponents?

Prime factorization with exponents expresses a composite number as the product of its prime factors raised to various powers. This article will teach you how to work with prime factorization with exponents.

For example, the number $$60$$ can be written as $$2^2 × 3 ×5$$. This means that $$60$$ is the product of $$2$$ raised to the power of $$2$$, multiplied by $$3$$ raised to the power of $$1$$ and multiplied by $$5$$ raised to the power of $$1$$. This is also known as the “expanded form” of prime factorization.

## A step-by-step guide toprime factorization with exponents

Prime factorization with exponents expresses a composite number (a positive integer greater than $$1$$ that is not a prime) as the product of prime numbers raised to specific powers. To make a prime factorization, you must break down a number into prime numbers which multiply by the original number.

A step-by-step guide to Prime Factorization with Exponents

1. Start by dividing the composite number by the smallest prime number that divides it.
2. Divide the result from step $$1$$ by the smallest prime number that divides it.
3. Repeat step $$2$$ until the quotient is no longer divisible by any prime numbers.
4. Write the prime factors and their exponents as a product of powers.

Example: Prime factorization of the number $$84$$.

1. $$84 ÷ 2 = 42$$. The prime factor $$2$$ is written as $$2^1$$.
2. $$42 ÷ 2 = 21$$. The prime factor $$2$$ is written as $$2^2$$.
3. $$21 ÷ 3 = 7$$. The prime factor $$3$$ is written as $$3^1$$.
4. $$7$$ is a prime number, so the prime factorization of $$84$$ is $$2^2 × 3^1 × 7^1$$. Therefore $$84 = 2^2×3×7$$.

Note: The prime factorization of a number is unique, which means that the order of the factors does not matter, but the exponents are important.

### Prime Factorization with Exponents – Example 1

Write the prime factorization with exponents.
$$32=……$$
Solution:
Divide $$32$$ by prime factors. The final quotient must be $$1$$.
The prime factorization of $$32$$ is $$4×8→4×4×2$$.
Write down the repeated factor $$(4)$$ with exponent, $$4^2×2$$.

### Prime Factorization with Exponents – Example 2

Write the prime factorization with exponents.
$$135=……$$
Solution:
Divide $$135$$ by prime factors. The final quotient must be $$1$$.
The prime factorization of $$135$$ is $$5×27→5×3×3×3$$
Write down the repeated factor $$(3)$$ with exponent, $$3^3×5$$.

## Exercises forPrime Factorization with Exponents

### Write the prime factorization with exponents.

1. $$\color{blue}{245}$$
2. $$\color{blue}{126}$$
3. $$\color{blue}{340}$$
1. $$\color{blue}{5\cdot 7^2}$$
2. $$\color{blue}{2\cdot 3^2\cdot7}$$
3. $$\color{blue}{2^2\cdot 5\cdot 17}$$

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