Each of these numbers is written as a product of prime factors. 539 = 11 \times 7^2 847 = 7 \times 11^2 Use this information to write \frac{539}{847} as a fraction in its simplest form,
The Correct Answer and Explanation is:
To simplify the fraction 539847\frac{539}{847}, we begin by expressing each number using its prime factorization:
539=11×72,847=7×112539 = 11 \times 7^2,\quad 847 = 7 \times 11^2
Now write the fraction using these factorizations:
539847=11×727×112\frac{539}{847} = \frac{11 \times 7^2}{7 \times 11^2}
Next, cancel the common prime factors in the numerator and denominator. Both terms include the primes 7 and 11. The numerator has one factor of 11 and two of 7, while the denominator has two 11s and one 7:
11×7×77×11×11\frac{11 \times 7 \times 7}{7 \times 11 \times 11}
Cancel one 7 and one 11 from top and bottom:
711\frac{7}{11}
So the fraction in its simplest form is 711\boxed{\frac{7}{11}}.
Explanation
Simplifying a fraction involves reducing it by canceling out any common factors between the numerator and the denominator. The most effective method begins with expressing each number as a product of prime numbers. This is known as prime factorization.
In this case, the numerator 539 breaks down into 11 multiplied by 7 twice, written as 11×7211 \times 7^2. The denominator 847 breaks down into 7 multiplied by 11 twice, or 7×1127 \times 11^2.
Once these expressions are substituted into the original fraction, we are able to see shared components. Both contain at least one 7 and one 11. By canceling a matching pair of each from the top and bottom, we eliminate redundancy. What remains after cancellation is a single 7 in the numerator and a single 11 in the denominator.
The result, 711\frac{7}{11}, is now in its lowest terms because 7 and 11 are both prime and share no common divisors other than 1. This demonstrates the power of prime factorization as a tool for simplifying fractions logically and systematically.
