Evaluating Expressions With Radicals Square Roots And Cube Roots

In this comprehensive guide, we will delve into the process of finding the values of various mathematical expressions involving square roots and cube roots. We will tackle three specific problems that showcase different techniques and approaches to simplify these expressions. This detailed exploration will not only provide the solutions but also offer insights into the underlying mathematical principles. Our main focus is to equip you with the knowledge and skills necessary to confidently solve similar problems in the future. In this article, we will solve three mathematical expressions. We will start by simplifying the expressions under the radicals, then apply the properties of radicals to further simplify them, and finally compute the numerical values. These steps will allow us to deal with mathematical problems effectively. We will also look at the different ways of solving them to help you understand better. Mathematical expressions are a fundamental part of mathematics, and understanding how to simplify and evaluate them is an essential skill for students and professionals alike. This guide aims to make the process clear and accessible, regardless of your mathematical background. We will use clear and concise language and avoid jargon whenever possible. So, let's begin our journey into the world of mathematical expressions and discover how to find their values.

(i) Finding the Value of ${ \sqrt{\frac{243}{1587}} }$

To determine the value of the expression ${ \sqrt{\frac{243}{1587}} }$, our initial step involves simplifying the fraction within the square root. This is achieved by identifying the greatest common divisor (GCD) of the numerator (243) and the denominator (1587). By finding the GCD, we can reduce the fraction to its simplest form, making it easier to work with. The process of simplifying fractions is a cornerstone of mathematical manipulation, especially when dealing with radicals and complex expressions. This not only makes the numbers more manageable but also reveals underlying mathematical structures that might not be immediately apparent. In this case, finding the GCD is crucial for simplifying the square root expression. Let's begin by finding the prime factorization of both 243 and 1587. This will allow us to systematically identify their common factors and ultimately determine the GCD. Once we have the simplified fraction, we can then apply the properties of square roots to further simplify the expression and arrive at the final value. This process of simplifying the fraction before dealing with the square root is a crucial step in efficiently solving this type of problem. The simplification also aids in reducing the risk of errors in the subsequent calculations, ensuring a more accurate final result.

Step 1: Simplify the Fraction

To simplify the fraction ${\frac{243}{1587}}$, we first find the prime factorization of both the numerator and the denominator. The prime factorization of 243 is ${3^5}$, meaning 243 can be expressed as 3 multiplied by itself five times (3 x 3 x 3 x 3 x 3). The prime factorization of 1587 is ${3^2 \times 177}$, which means 1587 can be expressed as the product of 3 squared (3 x 3) and 177. Now, we can rewrite the fraction using these prime factorizations:

${\frac{243}{1587} = \frac{3^5}{3^2 \times 177}}$

To simplify the fraction, we can divide both the numerator and the denominator by their common factors. In this case, both the numerator and the denominator have a common factor of ${3^2}$ (which is 9). Dividing both by ${3^2}$, we get:

${\frac{3^5}{3^2 \times 177} = \frac{3^5 \div 3^2}{(3^2 \times 177) \div 3^2} = \frac{3^3}{177}}$

So, the simplified fraction is ${\frac{3^3}{177}}$, which is equal to ${\frac{27}{177}}$. We can simplify this further by finding the prime factorization of 177, which is ${3 \times 59}$. Thus, we have:

${\frac{27}{177} = \frac{3^3}{3 \times 59}}$

Dividing both the numerator and the denominator by their common factor of 3, we get:

${\frac{3^3}{3 \times 59} = \frac{3^2}{59} = \frac{9}{59}}$

Therefore, the simplest form of the fraction ${\frac{243}{1587}}$ is ${\frac{9}{59}}$. This simplification process is crucial because it reduces the complexity of the numbers we are dealing with, making the subsequent square root calculation more manageable. By breaking down the numbers into their prime factors, we can identify and eliminate common factors, leading to a more streamlined and accurate result. This step highlights the importance of prime factorization in simplifying mathematical expressions and is a technique that can be applied in a wide range of mathematical problems.

Step 2: Evaluate the Square Root

Now that we have simplified the fraction ${\frac{243}{1587}}$ to ${\frac{9}{59}}$, we can proceed to evaluate the square root of this simplified fraction. The original expression was (\sqrt{\frac{243}{1587}}, so now we have (\sqrt{\frac{9}{59}},. To evaluate this, we can use the property of square roots that states the square root of a fraction is equal to the fraction of the square roots of the numerator and the denominator. In mathematical terms:

${\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}}$

Applying this property to our expression, we get:

${\sqrt{\frac{9}{59}} = \frac{\sqrt{9}}{\sqrt{59}}}$

The square root of 9 is 3, so we can simplify the numerator:

${\frac{\sqrt{9}}{\sqrt{59}} = \frac{3}{\sqrt{59}}}$

Now, to rationalize the denominator, we multiply both the numerator and the denominator by ${\sqrt{59}}$. This process eliminates the square root from the denominator, making the expression simpler and easier to work with. Rationalizing the denominator is a common practice in mathematics to present expressions in a standard form:

${\frac{3}{\sqrt{59}} = \frac{3}{\sqrt{59}} \times \frac{\sqrt{59}}{\sqrt{59}} = \frac{3\sqrt{59}}{59}}$

Therefore, the value of ${\sqrt{\frac{243}{1587}}}$ is ${\frac{3\sqrt{59}}{59}}$. This is the simplified form of the expression. This step demonstrates the application of fundamental properties of square roots and the technique of rationalizing the denominator. These are essential tools in simplifying and evaluating expressions involving radicals. The ability to manipulate radicals and fractions in this way is a crucial skill in algebra and other areas of mathematics. The final expression, ${\frac{3\sqrt{59}}{59}}$, is in its simplest form and provides the exact value of the original expression.

(ii) Finding the Value of ${ \sqrt[3]{\frac{33}{289}} }$

To find the value of the cube root expression ${ \sqrt[3]{\frac{33}{289}} }$, we follow a similar approach to the previous problem. First, we examine the fraction ${\frac{33}{289}}$ to see if it can be simplified. Simplifying the fraction before taking the cube root can make the calculation much easier. This involves finding the greatest common divisor (GCD) of the numerator (33) and the denominator (289). If the fraction can be simplified, we proceed with the simplified form; otherwise, we work with the original fraction. In this particular case, we will determine whether 33 and 289 share any common factors. If they do, dividing both by their GCD will simplify the fraction. This initial step is crucial for efficient calculation and minimizing the complexity of the expression under the cube root. By ensuring the fraction is in its simplest form, we reduce the chances of dealing with large numbers or complex factorizations later on. This preliminary simplification is a key strategy in handling radical expressions. Let's start by factoring 33 and 289 to see if we can simplify the fraction.

Step 1: Simplify the Fraction

Before we evaluate the cube root, let's simplify the fraction ${\frac{33}{289}}$. To do this, we need to find the prime factors of both the numerator (33) and the denominator (289). The prime factorization of 33 is ${3 \times 11}$. This means that 33 can be expressed as the product of the prime numbers 3 and 11. The prime factorization of 289 is ${17^2}$, which means 289 is the square of 17 (17 multiplied by itself). Now, we can rewrite the fraction using these prime factorizations:

${\frac{33}{289} = \frac{3 \times 11}{17^2}}$

Looking at the prime factorizations, we can see that 33 and 289 do not have any common factors. The numerator has prime factors 3 and 11, while the denominator has a prime factor of 17. Since there are no common factors between the numerator and the denominator, the fraction ${\frac{33}{289}}$ is already in its simplest form. This means we cannot simplify the fraction any further before taking the cube root. The inability to simplify the fraction at this stage indicates that we will need to work with the original numbers under the cube root. This understanding is important as it guides our next steps in the problem-solving process. In cases where simplification is not possible, we proceed directly to the evaluation of the radical, keeping in mind that the numbers may be larger or more complex to handle.

Step 2: Evaluate the Cube Root

Since the fraction ${\frac{33}{289}}$ is already in its simplest form, we can now evaluate the cube root of the fraction. The original expression is:

${\sqrt[3]{\frac{33}{289}}}$

Using the property of radicals that the cube root of a fraction is the fraction of the cube roots, we can rewrite the expression as:

${\sqrt[3]{\frac{33}{289}} = \frac{\sqrt[3]{33}}{\sqrt[3]{289}}}$

Now, we need to find the cube roots of 33 and 289. However, neither 33 nor 289 are perfect cubes, which means their cube roots are not integers. In such cases, we can leave the expression in this form, or we can try to simplify it further by looking for any perfect cube factors within 33 and 289. The prime factorization of 33 is ${3 \times 11}$, and there are no perfect cube factors in this factorization. The prime factorization of 289 is ${17^2}$, which is ${17 \times 17}$. Again, there are no perfect cube factors in this factorization. Therefore, we cannot simplify the cube roots any further. The expression ${\frac{\sqrt[3]{33}}{\sqrt[3]{289}}}$ is the simplified form of the cube root. To get a more precise value, we would typically use a calculator to find the cube roots of 33 and 289 and then divide. However, for the purpose of this problem, we can leave the answer in this form, which represents the exact value of the cube root of the fraction. This step highlights that not all radical expressions can be simplified to neat integers or simple fractions. In many cases, the best we can do is to express the value in terms of radicals. This understanding of the limits of simplification is an important part of working with radical expressions. The final expression represents the most simplified form achievable without resorting to approximations.

(iii) Finding the Value of ${ \sqrt{1083} \times \sqrt{363} }$

To find the value of the expression ${ \sqrt{1083} \times \sqrt{363} }$, we will use the property of square roots that allows us to multiply the numbers under the square root signs. This property states that ${\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}}$. Applying this property, we can combine the two square roots into a single square root, making the expression easier to manage. However, before we directly multiply 1083 and 363, it's often beneficial to simplify the numbers by finding their prime factorizations. This can reveal common factors that might simplify the expression under the square root. Simplifying the numbers before multiplying can lead to smaller numbers and easier calculations, reducing the chances of errors. This approach is particularly useful when dealing with large numbers or complex expressions. The initial step of finding prime factorizations is a key strategy in efficiently handling radical expressions. It allows us to identify and extract any perfect square factors, leading to a more simplified final result. This proactive simplification is a hallmark of effective mathematical problem-solving. Let's begin by finding the prime factorizations of 1083 and 363.

Step 1: Simplify by Prime Factorization

Before multiplying the square roots, let’s find the prime factorizations of 1083 and 363. The prime factorization of 1083 is ${3 \times 19^2}$. This means 1083 can be expressed as the product of 3 and the square of 19. The prime factorization of 363 is ${3 \times 11^2}$, which means 363 can be expressed as the product of 3 and the square of 11. Now, we can rewrite the original expression using these prime factorizations:

${\sqrt{1083} \times \sqrt{363} = \sqrt{3 \times 19^2} \times \sqrt{3 \times 11^2}}$

This step is crucial because it breaks down the numbers into their fundamental components, making it easier to identify and extract perfect squares from under the square root. By expressing the numbers in terms of their prime factors, we can simplify the expression before performing any further calculations. This simplification not only makes the numbers more manageable but also reveals the underlying structure of the expression, which can lead to a more elegant solution. The identification of perfect square factors (like ${19^2}$ and ${11^2}$) is key to the next step in simplifying the expression.

Step 2: Combine and Simplify the Square Roots

Now that we have the prime factorizations of 1083 and 363, we can combine the square roots and simplify the expression. We have:

${\sqrt{1083} \times \sqrt{363} = \sqrt{3 \times 19^2} \times \sqrt{3 \times 11^2}}$

Using the property ${\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}}$, we can combine the two square roots into one:

${\sqrt{3 \times 19^2} \times \sqrt{3 \times 11^2} = \sqrt{(3 \times 19^2) \times (3 \times 11^2)}}$

Now, we can simplify the expression under the square root by multiplying the factors:

${\sqrt{(3 \times 19^2) \times (3 \times 11^2)} = \sqrt{3^2 \times 19^2 \times 11^2}}$

We have ${3^2}$, ${19^2}$, and ${11^2}$ under the square root, which are all perfect squares. We can now take the square root of each of these factors. The square root of ${3^2}$ is 3, the square root of ${19^2}$ is 19, and the square root of ${11^2}$ is 11. Therefore, we can rewrite the expression as:

${\sqrt{3^2 \times 19^2 \times 11^2} = 3 \times 19 \times 11}$

Finally, we multiply these numbers together:

${3 \times 19 \times 11 = 3 \times 209 = 627}$

Therefore, the value of ${\sqrt{1083} \times \sqrt{363}}$ is 627. This process demonstrates the power of prime factorization in simplifying radical expressions. By breaking down the numbers into their prime factors, we were able to identify and extract perfect squares, leading to a straightforward calculation and a simplified final answer. The result is a clear integer, which is a testament to the effectiveness of this simplification strategy.

In this guide, we have successfully found the values of three mathematical expressions involving square roots and cube roots. We have demonstrated the importance of simplifying fractions and using prime factorization to make calculations easier. We also highlighted the properties of radicals that allow us to combine and simplify expressions. The first problem, finding the value of {\sqrt{\frac{243}{1587}}\, involved simplifying the fraction to \(\frac{9}{59}} and then rationalizing the denominator to arrive at the final answer of ${\frac{3\sqrt{59}}{59}}$. This problem emphasized the techniques of simplifying fractions and rationalizing denominators. The second problem, finding the value of {\sqrt[3]{\frac{33}{289}}\, showed us that not all expressions can be simplified to a neat integer or fraction, and the final answer was expressed in terms of cube roots, \(\frac{\sqrt[3]{33}}{\sqrt[3]{289}}\,. This problem illustrated the importance of recognizing when an expression is in its simplest form, even if it involves radicals. The third problem, finding the value of \(\sqrt{1083} \times \sqrt{363}}, demonstrated the use of prime factorization to simplify the numbers under the square roots and then combining the square roots to arrive at the integer answer of 627. This problem showcased the effectiveness of prime factorization in simplifying radical expressions and the power of recognizing perfect squares. These three problems provide a comprehensive overview of the techniques used to simplify and evaluate expressions involving radicals. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems involving square roots and cube roots. Remember, the key to success in mathematics is practice, so be sure to try these techniques on other similar problems to solidify your understanding.