Problem 36 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ 3 b y \sqrt{5 y}+4 \sqrt{5 b^{2} y^{3}}-2 \sqrt{5 b^{2} y^{3}} $$

Step-by-Step Solution

Answer

Question: What is the simplified form of the given algebraic expression: $$ 3 b y \sqrt{5 y}+4 \sqrt{5 b^{2} y^{3}}-2 \sqrt{5 b^{2} y^{3}} $$ Answer: The simplified form of the given algebraic expression is: $$ 3 b y \sqrt{5 y} + 2 \sqrt{5 b^{2} y^{3}} $$

Step 1: Identify the like terms

The algebraic expression given is: $$ 3 b y \sqrt{5 y}+4 \sqrt{5 b^{2} y^{3}}-2 \sqrt{5 b^{2} y^{3}} $$ First, we will identify the like terms in the expression. Notice that the second and third terms - \(4 \sqrt{5 b^{2} y^{3}}\) and \(-2 \sqrt{5 b^{2} y^{3}}\) - are like terms, since they have the same radical part \(\sqrt{5b^2y^3}\). We will combine these two terms in the next step.

Step 2: Combine like terms

Combining the like terms, we get: $$ 3 b y \sqrt{5 y} + (4-2) \sqrt{5 b^{2} y^{3}} $$ Solve the parentheses: $$ 3 b y \sqrt{5 y} + 2 \sqrt{5 b^{2} y^{3}} $$

Step 3: Write the final simplified expression

The expression is now simplified and cannot be combined further: $$ 3 b y \sqrt{5 y} + 2 \sqrt{5 b^{2} y^{3}} $$ So, the final simplified expression is: $$ 3 b y \sqrt{5 y} + 2 \sqrt{5 b^{2} y^{3}} $$

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a crucial step in simplifying algebraic expressions. When you see terms in an expression that have identical variable parts, such as the same variable or variables raised to the same power, these are termed 'like terms.' For example, if you have multiple terms in a radical expression that share the same radical part, then they can be combined by adding or subtracting their coefficients.

In the given exercise, we see a simplification that involves combining like terms with the same radical part. Even though the terms may look different because of their radical nature, you can treat the radicals as if they were variables and combine them accordingly. This is a handy tool to simplify complex expressions where you might have several terms under square roots or higher-order roots that are multiplied by different coefficients.

Radical Expressions

Dealing with radical expressions can be intimidating at first, but with the right approach, they can become manageable. A radical expression contains a root, such as a square root, cube root, etc. These expressions often require special attention when simplifying because you can only combine them with other radical expressions that have the same index and the same radicand, the value under the root symbol.

For instance, in our exercise, we have terms under square roots involving variables. To simplify such expressions, we ensure the radicands are identical before combining them. If they're not, we may need to factor or multiply the radicands to make them match or determine if common factors can be pulled out of the root to assist in the simplification process. Patience and practice are key in mastering radical expressions.

Algebraic Operations

Algebraic operations such as addition, subtraction, multiplication, and division are the building blocks of algebra. These operations help us to manipulate algebraic expressions and equations to reach simplified forms or solve for unknown variables. A good grasp of these operations becomes especially important when dealing with complex terms, such as those involving square roots and powers.

In the exercise, combining like terms is a form of algebraic addition. The process involves algebraic operations with radicals, which initially appears challenging, but adhering to the principle of combining like terms simplifies the process. Each algebraic operation follows specific rules that must be adhered to, ensuring accuracy and consistency in simplifying and solving expressions.

Square Roots

Square roots are a specific type of radical that are commonly encountered in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. Understanding how to work with square roots involves recognizing perfect squares, simplifying square roots through factorization, and knowing when to leave the expression in radical form.

In the exercise, square roots come into play within the radical expressions being combined. While simplifying these expressions, we are careful to only combine terms under the square root that are alike - those with the same number or expression after the radical symbol. It's crucial to note that not all square root expressions will simplify perfectly, and sometimes they may be left just as they are if they cannot be simplified further.