Simplifying Radicals A Detailed Solution For √45 + 2√20 - 15√5

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Introduction

In this comprehensive guide, we will delve into the process of simplifying radical expressions, focusing specifically on the expression $\sqrt{45}+2 \sqrt{20}-15 \sqrt{5}\$. Radicals, often represented by the square root symbol, can sometimes appear complex, but with the right techniques, they can be simplified into more manageable forms. This article aims to provide a step-by-step explanation of how to simplify this particular expression, while also offering broader insights into the principles of simplifying radicals in general. Understanding these principles is crucial for various areas of mathematics, from algebra to calculus, and mastering these techniques can significantly enhance your problem-solving skills.

Simplifying radicals involves breaking down the numbers under the radical sign into their prime factors and then extracting any perfect square factors. This process allows us to rewrite the expression in a more concise and understandable manner. For the expression at hand, we will first simplify each radical term individually before combining like terms. This approach not only makes the simplification process clearer but also reduces the chances of errors. By the end of this guide, you will have a solid understanding of how to tackle similar problems and a deeper appreciation for the elegance of mathematical simplification.

Breaking Down $\sqrt{45}\$

The first step in simplifying the expression $\sqrt{45}+2 \sqrt{20}-15 \sqrt{5}\∗∗istobreakdowntheterm∗∗$45** is to break down the term **\$\sqrt{45}\\. The number 45 can be factored into its prime factors, which are 3, 3, and 5. Mathematically, this can be represented as 45 = 3 × 3 × 5. The key to simplifying radicals is to identify perfect square factors. In this case, we have a pair of 3s, which can be written as 3². Therefore, we can rewrite $\sqrt{45}\$ as $\sqrt{3^2 × 5}\$.

Now, using the property of radicals that states $\sqrt{a × b} = \sqrt{a} × \sqrt{b}\$ (where a and b are non-negative numbers), we can separate the perfect square factor from the remaining factor. So, $\sqrt{3^2 × 5}\$ becomes $\sqrt{3^2} × \sqrt{5}\∗∗.Thesquarerootof32issimply3,thuswehave3$5**. The square root of 3² is simply 3, thus we have 3\$\sqrt{5}\\. This is the simplified form of $\sqrt{45}\∗∗.Understandinghowtobreakdownnumbersintotheirprimefactorsandidentifyperfectsquaresisfundamentaltosimplifyinganyradicalexpression.Thisprocessallowsustoextractthesquarerootsofperfectsquares,leavinguswithasimplifiedradicalterm.Inthenextstep,wewillapplythissametechniquetothenexttermintheexpression,2$20**. Understanding how to break down numbers into their prime factors and identify perfect squares is fundamental to simplifying any radical expression. This process allows us to extract the square roots of perfect squares, leaving us with a simplified radical term. In the next step, we will apply this same technique to the next term in the expression, 2\$\sqrt{20}\\.

Simplifying 2$\sqrt{20}\$

Having simplified $\sqrt45}\$ to 3$\sqrt{5}\$**, we now turn our attention to the second term in the original expression 2$\sqrt{20\∗∗.Thecoefficient2outsidetheradicalwillremainasitisfornow,andwewillfocusonsimplifyingtheradicalpart,$20**. The coefficient 2 outside the radical will remain as it is for now, and we will focus on simplifying the radical part, \$\sqrt{20}\\. To simplify $\sqrt{20}\$, we need to find the prime factors of 20. The prime factorization of 20 is 2 × 2 × 5, which can also be written as 2² × 5. This reveals that 20 has a perfect square factor, which is 2².

Therefore, we can rewrite $\sqrt{20}\∗∗as$22×5** as \$\sqrt{2^2 × 5}\\. Applying the property of radicals that allows us to separate the factors under the square root, we get $\sqrt{2^2} × \sqrt{5}\∗∗.Thesquarerootof22is2,sothissimplifiesto2$5**. The square root of 2² is 2, so this simplifies to 2\$\sqrt{5}\\. Now, we must not forget the coefficient 2 that was originally outside the radical. We multiply this coefficient by the simplified radical term, resulting in 2 × 2$\sqrt{5}\∗∗,whichequals4$5**, which equals 4\$\sqrt{5}\\. Thus, 2$\sqrt{20}\∗∗simplifiesto4$5** simplifies to 4\$\sqrt{5}\\. This step further demonstrates the importance of identifying perfect square factors within radicals to simplify them effectively. With two of the terms now simplified, we are one step closer to simplifying the entire expression. Next, we will look at the final term in the expression.

Analyzing 15$\sqrt{5}\$

The third and final term in our original expression $\sqrt{45}+2 \sqrt{20}-15 \sqrt{5}\∗∗is15$5** is 15\$\sqrt{5}\\. Unlike the previous terms, this term is already in its simplest form. The number under the radical, 5, is a prime number and has no perfect square factors other than 1. This means that $\sqrt{5}\$** cannot be simplified further. The coefficient 15 outside the radical remains as it is.

In this case, the term 15$\sqrt{5}\$** serves as a benchmark for the simplification process. It is a term with a radical that cannot be simplified further, and it will play a crucial role when we combine like terms in the final step. Recognizing when a radical is already in its simplest form is an important aspect of simplifying radical expressions. It prevents unnecessary steps and helps to focus on the terms that require simplification. Now that we have simplified all the individual terms, we are ready to combine them and arrive at the final simplified expression.

Combining Like Terms

Now that we have simplified each term individually, we can combine them to find the final simplified expression. Recall that we started with $\sqrt45}+2 \sqrt{20}-15 \sqrt{5}\∗∗.Wesimplified$45**. We simplified \$\sqrt{45}\\** to 3$\sqrt{5}\∗∗,2$20**, 2\$\sqrt{20}\\** to 4$\sqrt{5}\∗∗,andrecognizedthat15$5**, and recognized that 15\$\sqrt{5}\\** was already in its simplest form. Thus, our expression now looks like this 3$\sqrt{5\∗∗+4$5** + 4\$\sqrt{5}\\ - 15$\sqrt{5}\$**.

To combine these terms, we treat $\sqrt{5}\∗∗asacommonfactor,muchlikeavariableinanalgebraicexpression.Weaddandsubtractthecoefficientswhilekeepingtheradicalpartthesame.So,wehave(3+4−15)$5** as a common factor, much like a variable in an algebraic expression. We add and subtract the coefficients while keeping the radical part the same. So, we have (3 + 4 - 15)\$\sqrt{5}\\. Performing the arithmetic, 3 + 4 equals 7, and 7 - 15 equals -8. Therefore, the combined term is -8$\sqrt{5}\$. This is the final simplified form of the original expression.

Combining like terms is a fundamental step in simplifying expressions, whether they involve radicals or variables. It requires identifying terms that share a common factor and then performing the necessary arithmetic operations on their coefficients. In this case, by combining the simplified radical terms, we have successfully reduced the original expression to its simplest form, -8$\sqrt{5}\$**. This completes the simplification process, demonstrating the power of breaking down complex expressions into manageable parts and then combining them strategically.

Final Result

After meticulously breaking down each term in the expression $\sqrt45}+2 \sqrt{20}-15 \sqrt{5}\$** and combining like terms, we have arrived at the final simplified result **-8$\sqrt{5\$. This result encapsulates the entire process, from identifying perfect square factors within the radicals to performing arithmetic operations on the coefficients. The journey from the initial expression to this simplified form showcases the elegance and efficiency of mathematical simplification.

The simplified form, -8$\sqrt{5}\$**, is not only more concise but also easier to work with in further calculations or applications. It clearly represents the value of the original expression in its most basic terms. This process highlights the importance of mastering simplification techniques in mathematics, as they are essential for problem-solving and for gaining a deeper understanding of mathematical concepts. By following this step-by-step guide, one can confidently tackle similar problems and appreciate the beauty of simplifying complex expressions into their fundamental forms.

Conclusion

In conclusion, we have successfully simplified the expression $\sqrt{45}+2 \sqrt{20}-15 \sqrt{5}\∗∗to∗∗−8$5** to **-8\$\sqrt{5}\\ through a series of methodical steps. This process involved breaking down each radical term into its prime factors, identifying and extracting perfect square factors, and finally, combining like terms. The ability to simplify radical expressions is a crucial skill in mathematics, with applications spanning various fields, from algebra to calculus. This exercise not only demonstrates the specific steps required for this particular problem but also illustrates the general principles applicable to simplifying any radical expression.

The key takeaways from this guide include the importance of prime factorization, the identification of perfect square factors, and the strategic combination of like terms. By mastering these techniques, one can approach complex mathematical expressions with confidence and clarity. Moreover, this simplification process underscores the beauty of mathematics in its ability to reduce complex problems to their simplest, most elegant forms. The final result, -8$\sqrt{5}\$**, serves as a testament to the power of simplification and the importance of understanding the underlying principles of mathematical operations.