Problem 27 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 5 y \sqrt{8 y}+2 \sqrt{50 y^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Square Roots

Start by simplifying the square root expressions. Notice that both \( \sqrt{8y} \) and \( \sqrt{50y^3} \) can be simplified due to their factors.- For \( \sqrt{8y} \): 8 can be factorized into 4 and 2, so \( \sqrt{8y} = \sqrt{4 \cdot 2y} = 2\sqrt{2y} \).- For \( \sqrt{50y^3} \): 50 can be factorized into 25 and 2, and \( y^3 = y^2 \cdot y \), so \( \sqrt{50y^3} = \sqrt{25 \cdot 2y^2 \cdot y} = 5y\sqrt{2y} \).

Step 2: Substitute Simplified Square Roots

Substitute back the simplified expressions into the original equation:\[5y \sqrt{8y} + 2 \sqrt{50y^3} = 5y(2\sqrt{2y}) + 2(5y\sqrt{2y}) = 10y \sqrt{2y} + 10y \sqrt{2y}.\]

Step 3: Combine Like Terms

Now, combine the like terms from the expression obtained in Step 2. Each term contains \( 10y \sqrt{2y} \):\[10y \sqrt{2y} + 10y \sqrt{2y} = (10y + 10y) \sqrt{2y} = 20y \sqrt{2y}.\]

Key Concepts

Simplifying Square Roots

Simplifying square roots is a critical skill in algebra. To simplify a square root, the goal is to break it down into smaller, more manageable parts. This process often involves identifying perfect squares within the number under the square root. Here's how to do it:

• Identify any perfect square factors in the radicand (the number inside the square root).
• Rewrite the square root as a product of simpler square roots.
• Take the square root of the perfect square, which will come out of the square root.

For example, to simplify \( \sqrt{8y} \), recognize that 8 is \( 4 \times 2 \). Since 4 is a perfect square, it can be simplified to 2 when removed from the square root. Thus, \( \sqrt{8y} = 2\sqrt{2y} \). Similarly, \( \sqrt{50y^3} \) can be simplified by identifying the perfect square 25, as well as recognizing that \( y^3 = y^2 \cdot y \). This leads to \( \sqrt{50y^3} = 5y\sqrt{2y} \). Simplifying square roots helps streamline algebraic expressions, making calculations easier.

Combining Like Terms

Combining like terms is an essential technique in simplifying algebraic expressions. Like terms have the same variable raised to the same power, so they can be combined by addition or subtraction.

• Identify terms with identical variables and exponents.
• Add or subtract the coefficients of these terms.
• Maintain the shared variable part in the result.

In the expression \( 10y \sqrt{2y} + 10y \sqrt{2y} \), both terms are like terms because they share \( 10y \sqrt{2y} \). To combine them, simply add the coefficients: \( 10y + 10y = 20y \). This results in a simplified expression \( 20y \sqrt{2y} \). By combining like terms, the expression becomes more concise, facilitating further algebraic manipulations.

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots. Simplifying radical expressions is important to solve algebra problems successfully. A radical expression may consist of numbers, variables, or both under the root symbol.

• Step 1: Simplify each part of the radical if possible.
• Step 2: Look for like terms if the expression includes addition or subtraction.
• Step 3: Combine and simplify further if needed.

In our example, the initial expression \( 5y \sqrt{8y} + 2 \sqrt{50y^3} \) contains two separate radical terms. We first simplified each square root, then substituted to streamline the expression. Combining these into one final radical expression, \( 20y \sqrt{2y} \), lays the foundation for solving quadratic or complex algebraic problems. Understanding how to manage radical expressions ensures the ability to tackle diverse mathematical challenges.