Simplifying Algebraic Expressions A Step By Step Solution For 4 X^2 Sqrt 5 X^4 Cdot 3 Sqrt 5 X^8

In the realm of mathematics, particularly algebra, manipulating expressions to find equivalent forms is a fundamental skill. This process not only simplifies complex equations but also provides deeper insights into the relationships between variables and constants. This article delves into the intricacies of simplifying algebraic expressions, focusing on a specific example that involves exponents and radicals. We will dissect the problem step by step, providing a clear and concise explanation of the methodology and the underlying principles. By understanding these concepts, readers will be better equipped to tackle similar problems and enhance their algebraic proficiency.

Deconstructing the Expression: $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$

Our main task is to find an expression equivalent to $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$, given that $x \neq 0$. This problem requires a solid understanding of exponent and radical rules. Let's break down the expression piece by piece to simplify it effectively.

First, we have $4x^2$, a straightforward term where 4 is the coefficient and $x^2$ represents the variable $x$ raised to the power of 2. The next part, $\sqrt{5x^4}$, involves a square root. Remember, a square root is the same as raising to the power of $\frac{1}{2}$. Thus, $\sqrt{5x^4}$ can be rewritten as $(5x^4)^{\frac{1}{2}}$. Applying the power of a product rule, we get $5^{\frac{1}{2}} \cdot (x^4)^{\frac{1}{2}}$. Simplifying further, this becomes $\sqrt{5} \cdot x^2$. The final part of the expression is $3\sqrt{5x^8}$. Similarly, we rewrite $\sqrt{5x^8}$ as $(5x^8)^{\frac{1}{2}}$, which equals $5^{\frac{1}{2}} \cdot (x^8)^{\frac{1}{2}}$. Simplifying this gives us $3 \cdot \sqrt{5} \cdot x^4$. Now, we have all the components simplified and ready to be combined.

Step-by-Step Simplification

To begin, let's rewrite the original expression: $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$.

• Step 1: Simplify the radicals.
  • $\sqrt{5x^4} = \sqrt{5} \cdot \sqrt{x^4} = \sqrt{5} \cdot x^2$
  • $\sqrt{5x^8} = \sqrt{5} \cdot \sqrt{x^8} = \sqrt{5} \cdot x^4$
• Step 2: Substitute the simplified radicals back into the original expression.
  • $4x^2(\sqrt{5} \cdot x^2) \cdot 3(\sqrt{5} \cdot x^4)$
• Step 3: Rearrange and group like terms.
  • $(4 \cdot 3) \cdot (x^2 \cdot x^2 \cdot x^4) \cdot (\sqrt{5} \cdot \sqrt{5})$
• Step 4: Multiply the coefficients and use the exponent rule $x^a \cdot x^b = x^{a+b}$.
  • $12 \cdot x^{2+2+4} \cdot (\sqrt{5} \cdot \sqrt{5})$
  • $12 \cdot x^8 \cdot 5$
• Step 5: Multiply the remaining terms.
  • $60x^8$

Thus, the simplified expression is $60x^8$. This step-by-step approach ensures clarity and accuracy in the simplification process, making it easier to follow and understand the transformations applied.

Exponent Rules and Radical Simplification

Understanding exponent rules is crucial for simplifying algebraic expressions, especially those involving radicals. Exponents indicate the number of times a base is multiplied by itself, while radicals, such as square roots, represent the inverse operation of exponentiation. The connection between exponents and radicals allows us to rewrite radical expressions in exponential form and vice versa, making simplification easier.

Key Exponent Rules

• Product of Powers: When multiplying powers with the same base, add the exponents: $x^a \cdot x^b = x^{a+b}$. This rule was used when we combined the $x$ terms in our expression. For instance, $x^2 \cdot x^2 \cdot x^4$ became $x^{2+2+4} = x^8$.
• Power of a Power: When raising a power to another power, multiply the exponents: $(x^a)^b = x^{ab}$. This rule is essential for simplifying radicals. For example, $(x^4)^{\frac{1}{2}}$ becomes $x^{4 \cdot \frac{1}{2}} = x^2$.
• Power of a Product: When raising a product to a power, distribute the power to each factor: $(xy)^a = x^a y^a$. This rule allows us to separate terms within a radical. For instance, $\sqrt{5x^4}$ can be rewritten as $\sqrt{5} \cdot \sqrt{x^4}$.

Radical Simplification

Radicals can be simplified by understanding that $\sqrt[n]{x^m} = x^{\frac{m}{n}}$. This relationship is crucial for converting radicals into exponents and vice versa. In our problem, we used this principle to rewrite $\sqrt{5x^4}$ as $(5x^4)^{\frac{1}{2}}$ and then simplified it further. The goal is to remove perfect squares (or cubes, etc., depending on the root) from under the radical sign.

Applying the Rules to Our Problem

In the given expression, we applied these rules systematically. We converted the square roots into fractional exponents, used the power of a product rule to separate terms, and then applied the power of a power rule to simplify the exponents. By understanding these fundamental rules, we can efficiently manipulate and simplify complex algebraic expressions. For example, when we encountered $\sqrt{5x^8}$, we rewrote it as $(5x^8)^{\frac{1}{2}}$, then distributed the exponent to get $5^{\frac{1}{2}} \cdot (x^8)^{\frac{1}{2}}$, which simplified to $\sqrt{5} \cdot x^4$. This process highlights the power and flexibility of exponent rules in simplifying radical expressions.

Detailed Solution Walkthrough

To ensure a comprehensive understanding, let's walk through the solution again, highlighting each step and the reasoning behind it. The original expression is $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$.

Step 1: Simplify the Radicals

We begin by simplifying the radical terms. The square root of $5x^4$ can be written as $\sqrt{5x^4}$. Using the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we separate the terms: $\sqrt{5x^4} = \sqrt{5} \cdot \sqrt{x^4}$. The square root of $x^4$ is $x^2$ because $(x^2)^2 = x^4$. Therefore, $\sqrt{5x^4} = \sqrt{5} \cdot x^2$.

Similarly, we simplify $\sqrt{5x^8}$. Again, we separate the terms: $\sqrt{5x^8} = \sqrt{5} \cdot \sqrt{x^8}$. The square root of $x^8$ is $x^4$ because $(x^4)^2 = x^8$. Thus, $\sqrt{5x^8} = \sqrt{5} \cdot x^4$.

Step 2: Substitute Simplified Radicals

Now, we substitute the simplified radicals back into the original expression:

$4x^2(\sqrt{5} \cdot x^2) \cdot 3(\sqrt{5} \cdot x^4)$

Step 3: Rearrange and Group Like Terms

To make the multiplication clearer, we rearrange and group like terms together:

$(4 \cdot 3) \cdot (x^2 \cdot x^2 \cdot x^4) \cdot (\sqrt{5} \cdot \sqrt{5})$

This step helps to visually organize the terms before performing the multiplication.

Step 4: Multiply Coefficients and Apply Exponent Rules

We multiply the coefficients: $4 \cdot 3 = 12$. For the $x$ terms, we use the product of powers rule, which states that $x^a \cdot x^b = x^{a+b}$. Therefore, $x^2 \cdot x^2 \cdot x^4 = x^{2+2+4} = x^8$.

For the radical terms, $\sqrt{5} \cdot \sqrt{5} = 5$. So, the expression becomes:

$12 \cdot x^8 \cdot 5$

Step 5: Final Multiplication

Finally, we multiply the remaining terms: $12 \cdot 5 = 60$. Thus, the simplified expression is:

$60x^8$

This detailed walkthrough illustrates how each step is logically connected and how the exponent and radical rules are applied to reach the final simplified expression.

Identifying the Correct Answer

After simplifying the expression $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$, we arrived at $60x^8$. Now, let's compare this result with the given options to identify the correct answer.

The given options are:

• A. $12 x^{10} \sqrt{5}$
• B. $60 x^8$
• C. $35 x^{18}$
• D. $7 x^{10} \sqrt{5}$

Comparing Our Result

Our simplified expression, $60x^8$, matches option B exactly. Therefore, option B is the correct answer. The other options do not match our result, indicating they are incorrect.

• Option A, $12 x^{10} \sqrt{5}$, has a different coefficient and exponent for $x$, and it includes a $\sqrt{5}$ term, which is not present in our simplified expression.
• Option C, $35 x^{18}$, has a different coefficient and exponent for $x$.
• Option D, $7 x^{10} \sqrt{5}$, also has a different coefficient and exponent for $x$, and it includes a $\sqrt{5}$ term that is not in our simplified expression.

Thus, by carefully simplifying the original expression and comparing it with the provided options, we can confidently identify the correct answer as B. $60 x^8$. This process highlights the importance of accurate simplification and careful comparison in solving algebraic problems.

Common Mistakes to Avoid

When simplifying algebraic expressions involving exponents and radicals, several common mistakes can occur. Recognizing and avoiding these pitfalls is crucial for achieving accurate results. Let's explore some of these common errors and how to prevent them.

Incorrectly Applying Exponent Rules

One of the most frequent mistakes is misapplying exponent rules. For example, the rule $x^a \cdot x^b = x^{a+b}$ is often confused with $(x^a)^b = x^{ab}$. In the first case, you add the exponents when multiplying powers with the same base. In the second case, you multiply the exponents when raising a power to another power. Confusing these rules can lead to incorrect simplification. To avoid this, always double-check which rule applies to the situation.

Misunderstanding Radical Simplification

Another common mistake is incorrectly simplifying radicals. Remember that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, but $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$. Also, when simplifying $\sqrt{x^n}$, if $n$ is even, the result is $|x^{\frac{n}{2}}|$, and if $n$ is odd, the result is $|x^{\frac{n-1}{2}}| \sqrt{x}$. Forgetting these nuances can lead to errors. In our problem, correctly simplifying $\sqrt{5x^4}$ to $\sqrt{5} \cdot x^2$ is essential.

Errors in Arithmetic

Simple arithmetic errors can also derail the simplification process. For instance, multiplying coefficients incorrectly or adding exponents wrongly can lead to an incorrect final answer. It's always a good practice to double-check each arithmetic operation, especially when dealing with multiple steps.

Forgetting the Coefficient

Sometimes, students might forget to multiply the coefficients when simplifying. In our expression, we had $4x^2 \cdot 3\sqrt{5x^8}$. The coefficients 4 and 3 need to be multiplied along with the other terms. Forgetting to do so will result in an incorrect coefficient in the final answer.

Incorrectly Distributing Exponents

When dealing with expressions like $(xy)^a$, remember to distribute the exponent to both $x$ and $y$, resulting in $x^a y^a$. A common mistake is to apply the exponent only to one term. In our problem, correctly applying the power of a product rule is crucial for simplifying radicals.

By being mindful of these common mistakes and taking the time to double-check each step, you can significantly improve your accuracy in simplifying algebraic expressions.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions involving exponents and radicals is a crucial skill in mathematics. By understanding and applying the fundamental rules of exponents and radicals, we can effectively manipulate complex expressions into simpler, equivalent forms. This process not only aids in solving equations but also enhances our understanding of mathematical relationships.

Throughout this article, we have dissected the expression $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$, providing a step-by-step guide to its simplification. We emphasized the importance of exponent rules, such as the product of powers, power of a power, and power of a product, as well as the principles of radical simplification. By converting radicals to exponential form and vice versa, we were able to efficiently simplify the expression.

We also highlighted common mistakes to avoid, such as misapplying exponent rules, incorrectly simplifying radicals, arithmetic errors, forgetting coefficients, and incorrectly distributing exponents. By being aware of these potential pitfalls, students can improve their accuracy and confidence in algebraic manipulations.

The correct answer to the problem is B. $60 x^8$. This was achieved by systematically simplifying the expression, applying the relevant rules, and carefully comparing the result with the given options.

Mastering algebraic simplification requires practice and a thorough understanding of the underlying principles. By consistently applying these techniques, students can develop proficiency in this essential mathematical skill, paving the way for success in more advanced topics. This comprehensive approach ensures that readers gain not only the ability to solve specific problems but also a deeper appreciation for the elegance and power of algebraic manipulation.

By focusing on clear explanations, detailed walkthroughs, and practical tips, this article aims to empower students to tackle algebraic expressions with confidence and precision. The journey to mastering algebra is ongoing, and each problem solved is a step forward in this rewarding endeavor.