Write 168 as the product of prime factors (index form)
The Correct Answer and Explanation is:
Correct Answer:
168 = 2 × 3 × 7
Explanation
To express the number 168 as a product of its prime factors in index form, the process of prime factorization is used. This method involves breaking the number down into factors that are all prime numbers.
The number 168 is even, meaning it is divisible by 2, the smallest prime number. Dividing 168 by 2 gives 84. Since 84 is also even, it can be divided by 2 again, giving 42. Continuing, 42 divided by 2 gives 21. At this point, 21 is no longer even and cannot be divided by 2 again.
The next smallest prime number after 2 is 3. Dividing 21 by 3 gives 7. Finally, 7 is itself a prime number and cannot be broken down further.
The complete prime factorization of 168 results in the prime numbers 2, 2, 2, 3, and 7. This can be written as a multiplication of these factors:
168 = 2 × 2 × 2 × 3 × 7.
To write this in index form, similar prime numbers are grouped together and expressed using exponents. The number 2 appears three times, so it is expressed as 2³. The numbers 3 and 7 each appear once, so they are written as 3¹ and 7¹, although the exponent 1 is usually omitted in final form. Therefore, the product in index form is:
168 = 2³ × 3 × 7.
Prime factorization plays an important role in various areas of mathematics, including simplifying fractions, finding greatest common factors, and solving problems involving divisibility. It reveals the building blocks of a number and provides insight into its structure, making it a valuable technique in both arithmetic and algebraic calculations.
