Simplifying Square Root Of 48 Using Prime Factorization A Detailed Explanation

In the realm of mathematics, simplifying square roots is a fundamental skill, especially when dealing with numbers that aren't perfect squares. One powerful technique to achieve this is through prime factorization. Prime factorization involves breaking down a number into its prime factors, which are numbers divisible only by 1 and themselves. This method allows us to identify pairs of factors that can be extracted from the square root, leading to the simplest form. In this comprehensive explanation, we will delve into the process of simplifying $\sqrt{48}\$ using prime factorization, highlighting the correct steps and elucidating why certain approaches are more effective than others.

When tackling the simplification of $\sqrt{48}\$ using prime factorization, the primary objective is to express 48 as a product of its prime factors. To initiate this process, one might begin by observing that 48 is an even number, hence divisible by 2. Dividing 48 by 2 yields 24, which is also an even number and thus divisible by 2 again. This division results in 12, which can be further divided by 2 to obtain 6. Finally, 6 can be divided by 2 to get 3, a prime number. Therefore, the prime factorization of 48 can be expressed as 2 * 2 * 2 * 2 * 3, or equivalently, 2^4 * 3. This prime factorization lays the groundwork for simplifying the square root.

With the prime factorization of 48 at hand, the next step involves expressing $\sqrt{48}\$ as $\sqrt{2^4 * 3}\,. The beauty of this representation lies in the properties of square roots. Recall that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}\$ holds true for non-negative numbers a and b. Applying this property, we can rewrite $\sqrt{2^4 * 3}\$ as $\sqrt{2^4} * \sqrt{3}\,. Now, the key observation is that 2^4 is a perfect square, specifically (2²⁾2. Therefore, $\sqrt{2^4}\$ simplifies to 2^2, which equals 4. Consequently, the expression becomes 4 * $\sqrt{3}\$ or 4$\sqrt{3}\$ This final form represents the simplest form of $\sqrt{48}\$ achieved through the method of prime factorization.

When simplifying square roots, it's crucial to follow the correct procedure to arrive at the simplest form. Often, there are alternative approaches that may seem intuitive but can lead to incorrect results or non-simplified forms. Let's examine some incorrect methods and understand why they fall short.

Consider the expression $\sqrt{48}\,. One might be tempted to factor 48 as 4 * 12 and rewrite the square root as $\sqrt{4 * 12}\$ This approach, while not inherently wrong, does not lead to the simplest form directly. Applying the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}\$ we get $\sqrt{4} * \sqrt{12}\$ which simplifies to 2$\sqrt{12}\$ While this step is mathematically correct, the expression 2$\sqrt{12}\$ is not in its simplest form because $\sqrt{12}\$ can be further simplified. The number 12 can be factored as 4 * 3, and $\sqrt{12}\$ can be written as $\sqrt{4 * 3}\$ which equals 2$\sqrt{3}\$ Substituting this back into the expression, we get 2 * 2$\sqrt{3}\$ which simplifies to 4$\sqrt{3}\$ This demonstrates that while factoring out 4 initially was a valid step, it required further simplification to reach the ultimate simplest form.

Another common mistake is to stop at an intermediate step without fully factoring the number under the square root. For instance, one might factor 48 as 2 * 2 * 2 * 2 * 3, which is the correct prime factorization. However, an error occurs if this is prematurely written as 2$\sqrt{12}\$ as in option A. Although 2 * 2 * 2 * 2 * 3 is indeed the prime factorization, the simplification to 2$\sqrt{12}\$ misses the crucial step of extracting all possible perfect square factors. The correct approach involves grouping the factors into pairs. Here, the pair 2 * 2 appears twice, which means 2^4 can be extracted from the square root as 2^2 = 4. The remaining factor, 3, stays under the square root, resulting in the simplest form 4$\sqrt{3}\$ Thus, the error lies in not fully utilizing the prime factorization to identify and extract all perfect square factors.

Additionally, option B illustrates a similar but distinct error. It correctly starts by factoring 48 as 4 * 12 but then incorrectly concludes that this is the simplest form. While $\sqrt{4}\$ simplifies to 2, the $\sqrt{12}\$ still contains a perfect square factor (4). This oversight prevents the expression from being fully simplified. The correct procedure requires further breaking down $\sqrt{12}\$ into $\sqrt{4 * 3}\$ and simplifying to 2$\sqrt{3}\$ Ultimately, this demonstrates that the simplest form is achieved only when all perfect square factors have been extracted from the square root.

The correct approach to simplifying $\sqrt{48}\$ using prime factorization involves a systematic breakdown of 48 into its prime factors, followed by the extraction of perfect squares. This method ensures that the square root is reduced to its simplest form, leaving no room for further simplification. Let's walk through the steps meticulously.

As previously established, the prime factorization of 48 is 2 * 2 * 2 * 2 * 3, which can also be written as 2^4 * 3. This factorization provides the foundation for simplifying $\sqrt{48}\$ Substituting the prime factorization into the square root, we get $\sqrt{2^4 * 3}\$ The next crucial step is to utilize the property of square roots that allows us to separate factors under the radical sign. Specifically, $\sqrt{a * b} = \sqrt{a} * \sqrt{b}\$ for non-negative numbers a and b. Applying this property, we rewrite $\sqrt{2^4 * 3}\$ as $\sqrt{2^4} * \sqrt{3}\$ This separation is pivotal because it isolates the perfect square factor, 2^4, which can be easily simplified.

Now, we focus on simplifying $\sqrt{2^4}\$ Recognizing that 2^4 is (2²⁾2, we can directly simplify its square root. Recall that $\sqrt{x^2} = |x|$, but since we are dealing with positive numbers in this context, we can simply say $\sqrt{x^2} = x$. Thus, $\sqrt{2^4}\$ becomes $\sqrt{(2²⁾2}\$ which simplifies to 2^2, or 4. Substituting this result back into our expression, we now have 4 * $\sqrt{3}\$ or 4$\sqrt{3}\$ This is the simplified form of $\sqrt{48}\$ achieved through the correct application of prime factorization and square root properties.

The expression 4$\sqrt{3}\$ is in its simplest form because the number under the square root, 3, is a prime number and has no perfect square factors other than 1. This means that no further simplification is possible. The coefficient 4 is an integer, and the radical $\sqrt{3}\$ is in its most reduced state. This meticulous process of prime factorization and simplification ensures that we have arrived at the ultimate simplest form of the given square root.

In conclusion, the correct calculation that uses prime factorization to write $\sqrt{48}\$ in simplest form involves breaking down 48 into its prime factors (2 * 2 * 2 * 2 * 3), expressing $\sqrt{48}\$ as $\sqrt{2^4 * 3}\$ separating the square root into $\sqrt{2^4} * \sqrt{3}\$ simplifying $\sqrt{2^4}\$ to 4, and finally writing the simplest form as 4$\sqrt{3}\$ This method exemplifies the power and precision of prime factorization in simplifying square roots, ensuring that the final expression is indeed in its most reduced state.

In summary, simplifying square roots through prime factorization is a meticulous yet powerful technique. The key lies in accurately breaking down the number into its prime factors and then extracting pairs to simplify the radical. The correct calculation, as demonstrated, meticulously follows this process, ensuring that $\sqrt{48}\$ is simplified to 4$\sqrt{3}\$ This detailed explanation underscores the importance of understanding prime factorization and its application in simplifying square roots within the broader field of mathematics.