Simplify: (7√x + 5)(6√x - 9) - Radicals & Variables

In the realm of mathematics, simplifying expressions involving radicals and variables is a fundamental skill. This guide will walk you through the process of simplifying the expression (7√x + 5)(6√x - 9), assuming that all variables represent positive real numbers. We will break down each step, ensuring clarity and understanding. Whether you are a student tackling algebra or simply looking to refresh your math skills, this article provides a detailed, step-by-step approach to handling such expressions. Let’s dive in and make math a little less daunting!

Understanding the Basics

Before we tackle the main problem, let's ensure we have a solid understanding of the basic principles involved. When simplifying expressions with radicals, remember these key points:

• Radicals: A radical is a symbol (√) that indicates the root of a number. For example, √x represents the square root of x.
• Variables: In our case, 'x' represents a variable, which is a symbol for a number we don't yet know. We assume x is a positive real number.
• Distribution: The distributive property is crucial for multiplying expressions like the one we're dealing with. It states that a(b + c) = ab + ac.
• Combining Like Terms: After multiplying, we simplify by combining terms that have the same variable and radical parts.

The Distributive Property

The distributive property is the cornerstone of multiplying expressions like (7√x + 5)(6√x - 9). This property allows us to multiply each term in the first expression by each term in the second expression. Think of it like this: every term gets a chance to "distribute" itself across the other expression. The distributive property is essential for expanding and simplifying algebraic expressions. It ensures that each term within a set of parentheses is properly multiplied by a term outside the parentheses, leading to a correct expansion. This is especially vital when dealing with expressions that contain multiple terms and variables, as it allows us to break down complex multiplications into simpler, manageable steps. Without a solid understanding of the distributive property, tackling more advanced algebraic problems becomes significantly more challenging.

Simplifying Radicals

Simplifying radicals is a critical skill in algebra. A radical is in its simplest form when the radicand (the number under the radical sign) has no perfect square factors other than 1. For example, √8 can be simplified to 2√2 because 8 has a perfect square factor of 4. Simplifying radicals makes expressions easier to understand and work with. It allows for more straightforward comparisons and combinations of terms. When simplifying, always look for the largest perfect square factor of the radicand and extract its square root. This not only reduces the complexity of the expression but also ensures that your final answer is in its most concise and readable form. Mastering the simplification of radicals is a stepping stone to more advanced topics in algebra and calculus, providing a foundation for handling complex mathematical problems.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. Like terms are those that have the same variable raised to the same power. For example, 3x and 5x are like terms because they both contain the variable x raised to the power of 1. Similarly, 2√x and 7√x are like terms because they both contain the square root of x. Combining like terms involves adding or subtracting their coefficients (the numbers in front of the variables) while keeping the variable part the same. This process reduces the number of terms in an expression and makes it easier to understand and work with. It's essential to pay close attention to the signs of the coefficients and to ensure that you're only combining terms that are truly alike. Mastering this skill is crucial for simplifying more complex expressions and solving algebraic equations efficiently.

Step-by-Step Solution

Let’s apply these principles to simplify the given expression: (7√x + 5)(6√x - 9).

Step 1: Apply the Distributive Property

We multiply each term in the first expression by each term in the second expression:

(7√x + 5)(6√x - 9) = (7√x)(6√x) + (7√x)(-9) + (5)(6√x) + (5)(-9)

Step 2: Perform the Multiplication

Now, let's perform each multiplication:

• (7√x)(6√x) = 42x
• (7√x)(-9) = -63√x
• (5)(6√x) = 30√x
• (5)(-9) = -45

So, our expression becomes:

42x - 63√x + 30√x - 45

Step 3: Combine Like Terms

We combine the terms that contain √x:

-63√x + 30√x = -33√x

Now, our expression is:

42x - 33√x - 45

Final Answer

Thus, the simplified form of the expression (7√x + 5)(6√x - 9) is:

42x - 33√x - 45

Common Mistakes to Avoid

When simplifying radical expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy.

Mistake 1: Incorrectly Applying the Distributive Property

One frequent error is not correctly distributing each term in the first expression to every term in the second expression. For example, forgetting to multiply the '5' by both '6√x' and '-9' in our original problem (7√x + 5)(6√x - 9). Always double-check that each term is properly multiplied. A systematic approach, such as writing out each multiplication explicitly, can help prevent this mistake.

Mistake 2: Improperly Multiplying Radicals

Another common mistake involves multiplying radicals incorrectly. Remember that (√x)(√x) = x, not √x². Ensure you understand how radicals interact when multiplied. For instance, in our problem, (7√x)(6√x) becomes 42x because √x times √x equals x. Review the rules of radical multiplication to avoid this error.

Mistake 3: Not Combining Like Terms Correctly

Failing to combine like terms or combining unlike terms is another pitfall. Only terms with the exact same radical part can be combined. For example, -63√x and 30√x can be combined because they both contain √x. However, 42x cannot be combined with -33√x because they are not like terms. Carefully examine each term to ensure you're only combining those that are alike.

Mistake 4: Forgetting to Simplify Radicals

Sometimes, students forget to simplify radicals before or after performing other operations. Always look for perfect square factors within the radical and simplify them. For example, if you encounter √8, simplify it to 2√2 before proceeding. Simplifying radicals at each step ensures the final answer is in its most reduced form.

Mistake 5: Arithmetic Errors

Simple arithmetic errors, such as incorrect addition or subtraction, can also lead to wrong answers. Double-check your calculations, especially when dealing with negative numbers. It's easy to make a mistake when subtracting a negative number or adding numbers with different signs. Using a calculator for basic arithmetic can help reduce these errors.

Conclusion

Simplifying expressions involving radicals and variables requires a clear understanding of basic principles such as the distributive property, simplifying radicals, and combining like terms. By following the step-by-step solution outlined above and avoiding common mistakes, you can confidently tackle these types of problems. Practice is key to mastering these skills, so keep working at it! Remember, math is like any other skill; the more you practice, the better you become.

For further learning and practice on similar mathematical concepts, consider visiting Khan Academy's Algebra Section. This resource offers numerous exercises and video tutorials to enhance your understanding and skills.