Finding The Equivalent Expression Of 3√1089 A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. This article will delve into the process of finding an expression equivalent to 33{sqrt{1089}}$. We'll explore various simplification techniques, focusing on identifying perfect squares within the square root. By breaking down the number 1089 and applying the properties of square roots, we'll arrive at the simplified form and discuss why each of the provided options is either equivalent or not. This exploration will not only help in solving this specific problem but also provide a solid foundation for tackling similar mathematical challenges. Understanding the principles behind simplifying radical expressions is crucial for success in algebra and beyond. So, let's embark on this mathematical journey to unlock the equivalent expression for 33{sqrt{1089}}$.

Deconstructing the Expression: 3√1089

To effectively tackle this problem, our first key step involves understanding the structure of the given expression, 33sqrt{1089}}$. This expression consists of two main components the coefficient 3 and the square root of 1089. The coefficient 3 is a simple integer, but the square root of 1089 requires further investigation. To simplify this expression, we need to determine if 1089 is a perfect square or if it can be factored into a product containing a perfect square. This is where our knowledge of number theory and factorization comes into play. We will explore methods such as prime factorization to break down 1089 into its constituent factors. By identifying any perfect square factors, we can extract them from the square root, thus simplifying the overall expression. This process is crucial because it allows us to rewrite the expression in a more manageable form, making it easier to compare with the given options. Remember, the goal is to find an expression that has the same value as 33{sqrt{1089}$, and simplifying the square root is the most direct path to achieving this.

Prime Factorization of 1089: Unveiling the Perfect Square

The next critical step in simplifying 33{sqrt{1089}}$ is to perform the prime factorization of 1089. Prime factorization is the process of breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. This method is particularly useful when dealing with square roots because it helps us identify perfect square factors. Let's begin by dividing 1089 by the smallest prime number, 2. Since 1089 is an odd number, it is not divisible by 2. Next, we try the next prime number, 3. 1089 divided by 3 gives us 363. So, we have 1089 = 3 * 363. Now, we need to factor 363. 363 is also divisible by 3, giving us 121. So, 363 = 3 * 121. Now our factorization looks like this: 1089 = 3 * 3 * 121. Notice that 121 is a well-known perfect square, as it is 11 squared (11 * 11). Therefore, we can write 121 as 11^2. Putting it all together, the prime factorization of 1089 is 3 * 3 * 11 * 11, which can be written as 3^2 * 11^2. This prime factorization is a key breakthrough, as it clearly shows that 1089 is composed of perfect square factors. Understanding this factorization allows us to simplify the square root and ultimately find the equivalent expression.

Simplifying the Radical: Extracting Perfect Squares

Now that we have the prime factorization of 1089 as 32∗1123^2 * 11^2, we can proceed with simplifying the square root in the expression 33sqrt{1089}}$. Recall that the square root of a product is equal to the product of the square roots. In other words, ${sqrt{a * b}}$ = sqrta{sqrt{a}} * sqrtb{sqrt{b}}$. Applying this property to our expression, we can rewrite ${sqrt{1089}}$ as ${sqrt{3^2 * 11^2}}$. This can further be broken down into ${sqrt{3^2}}$ * sqrt112{sqrt{11^2}}$. The square root of a number squared is simply the number itself. Therefore, ${sqrt{3^2}}$ equals 3, and ${sqrt{11^2}}$ equals 11. Substituting these values back into our expression, we have ${sqrt{1089}}$ = 3 * 11, which equals 33. Now, we can substitute this simplified value back into the original expression 33{sqrt{1089}$ becomes 3 * 33. Multiplying these two numbers together, we get 99. So, the simplified value of the expression 33{sqrt{1089}}$ is 99. This simplification is a crucial step in identifying the equivalent expression among the given options. By extracting the perfect squares from the radical, we've transformed the original expression into a simple integer, making it much easier to compare with the other options.

Evaluating the Options: Identifying the Equivalent Expression

With the simplified value of the original expression 33{sqrt{1089}}$ determined to be 99, we can now evaluate the provided options to identify the equivalent expression. Each option presents a different form, and we need to simplify each one to see if it also equals 99. Let's examine each option in detail:

  1. 33sqrt{32(11)2}}$ This expression can be simplified by recognizing that 323^2 is 9 and 11211^2 is 121. So, the expression becomes 33{sqrt{9 * 121}$. As we discussed earlier, the square root of a product is the product of the square roots, so we can rewrite this as $3 * sqrt9{sqrt{9}} * sqrt121{sqrt{121}}$. The square root of 9 is 3, and the square root of 121 is 11. Therefore, the expression simplifies to 3 * 3 * 11, which equals 99. This option matches our simplified value of the original expression.

  2. 33{sqrt{121}}$: The square root of 121 is 11, so this expression simplifies to 3 * 11, which equals 33. This is not equal to 99, so this option is not equivalent.

  3. 33{sqrt{14^2}}$: The square root of 14214^2 is 14, so this expression simplifies to 3 * 14, which equals 42. This is not equal to 99, so this option is not equivalent.

  4. 33sqrt{11^3}}$ 11311^3 is 11 * 11 * 11, which equals 1331. So, this expression is 33{sqrt{1331}$. We can rewrite ${sqrt{1331}}$ as ${sqrt{11^2 * 11}}$, which simplifies to 11sqrt11{sqrt{11}}.Therefore,theexpressionbecomes3∗11. Therefore, the expression becomes 3 * 11{sqrt{11}}$, which equals 33sqrt11{sqrt{11}}$. This is not equal to 99, so this option is not equivalent.

By evaluating each option, we can clearly see that only the first option, 33{sqrt{32(11)2}}$, simplifies to 99, which is the same as the simplified value of the original expression. Therefore, this is the equivalent expression we were looking for.

Conclusion: The Equivalent Expression Unveiled

In conclusion, after a thorough analysis and simplification process, we have successfully identified the expression equivalent to 33{sqrt{1089}}$. By breaking down the number 1089 into its prime factors, extracting perfect squares from the radical, and simplifying the resulting expression, we determined that 33{sqrt{1089}}$ equals 99. Subsequently, we evaluated each of the provided options and found that only 33{sqrt{32(11)2}}$ also simplifies to 99. Therefore, 33{sqrt{32(11)2}}$ is the expression equivalent to 33{sqrt{1089}}$. This exercise highlights the importance of understanding prime factorization, the properties of square roots, and the techniques for simplifying radical expressions. These skills are fundamental in mathematics and are essential for solving a wide range of problems. By mastering these concepts, students can confidently tackle algebraic challenges and achieve success in their mathematical endeavors. This journey through simplification not only provides the answer to a specific question but also reinforces the broader principles of mathematical problem-solving.