How to Solve 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 to prime 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 that 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.

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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, \(3^3×5\).

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Exercises for Prime 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. \(\color{blue}{2^2\cdot 5\cdot 17}\)