Problem 27 - Intermediate Algebra

Question

Add or subtract. $$ 5 y \sqrt{8 y}+2 \sqrt{50 y^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(20y\sqrt{2y}\).

Step 1: Simplify each Term

First, we need to simplify the square root parts of each term. For the first term, we have:\[\sqrt{8y} = \sqrt{4 \cdot 2 \cdot y} = \sqrt{4} \times \sqrt{2y} = 2\sqrt{2y}.\]So, the first term becomes:\[5y \times 2\sqrt{2y} = 10y\sqrt{2y}.\]For the second term, we have:\[\sqrt{50y^3} = \sqrt{25 \cdot 2 \cdot y^2 \cdot y} = \sqrt{25} \times \sqrt{y^2} \times \sqrt{2y} = 5y\sqrt{2y}.\]So, the second term becomes:\[2 \times 5y\sqrt{2y} = 10y\sqrt{2y}.\]

Step 2: Add Like Terms

Now, both simplified expressions are like terms and can be combined.The expression becomes:\[10y\sqrt{2y} + 10y\sqrt{2y}.\]Since these are like terms, add them together:\[20y\sqrt{2y}.\]

Step 3: Conclusion: Final Expression

After simplifying and combining like terms, the final result of the given expression is:\[20y\sqrt{2y}.\]

Key Concepts

Square Roots and Radicals

Understanding square roots and radicals is key to simplifying algebraic expressions. A square root, denoted by the symbol \( \sqrt{} \), represents a number that, when multiplied by itself, gives the original number. Radicals refer to expressions that include a root symbol. Simplifying a square root involves breaking it down into its simplest form. For example, in the expression \( \sqrt{8y} \), it's rewritable as \( \sqrt{4 \times 2y} \). Since \( \sqrt{4} \) equals 2, this expression simplifies to \( 2\sqrt{2y} \). It's important to identify perfect squares within a square root, as it allows for easy simplification. Similarly, reducing \( \sqrt{50y^3} \) involves recognizing it as \( \sqrt{25 \times 2y^2 \times y} \). Again, \( \sqrt{25} \) results in 5 and \( \sqrt{y^2} \) provides \( y \), leading to the simplification \( 5y\sqrt{2y} \). To successfully simplify square roots and radicals, identify and extract perfect square factors. This will make your calculations easier and faster. Look for numbers and variables that appear in even powers, as these can help transform a complex radical into a more manageable form.

Adding and Subtracting Like Terms

Once radical expressions are simplified, combining like terms is the next logical step. Like terms have the same variables and powers, as well as identical radical parts. This implies that the numerical coefficients can be added or subtracted easily. In the exercise, both terms simplify to \( 10y\sqrt{2y} \). Since they share the exact variable structure and radical component, they are indeed like terms. When handling expressions with like terms, it's helpful to:

• Verify that each term has the same variable and corresponding powers.
• Ensure the radical portion matches completely between the terms.

Once you establish these criteria, simply combine the coefficients, which here results in \( 20y\sqrt{2y} \). By understanding how to identify and manage like terms, you'll be able to streamline expressions involving radicals.

Simplification Techniques

The art of simplification drives much of algebraic manipulation, ensuring expressions are expressed in the simplest form possible. Simplification techniques are fundamental for a clean and clear final result. Ensuring each step of simplification is guided by principles will make these tasks much easier.Begin simplification by:

• Breaking down complex radicals into their most basic components.
• Fully simplifying inside of any roots wherever possible.
• Carefully combining like terms by matching variable powers and radical components.

It's not just about reaching the answer quickly, but about understanding the process. For example, by fully simplifying \( 5y \sqrt{8y} \) and \( 2\sqrt{50y^3} \), you can seamlessly combine them later. Every time we simplify, we're making the expression easier to handle and interpret.Remember, algebra is a language. The simpler the form, the clearer the message. Use these techniques not only to get the right answer but to present mathematics clearly and concisely.