Problem 10 - Prealgebra

Question

Combine by applying the distributive property. Assume all variables represent positive numbers. $$7 \sqrt{y}+\sqrt{y}+2 \sqrt{y}$$

Step-by-Step Solution

Answer

Combine the terms to get \(10 \sqrt{y}\).

Step 1: Identifying Like Terms

We start by noticing that all the terms involve the expression \(\sqrt{y}\). This makes them like terms, which we can combine using the distributive property.

Step 2: Factoring Out the Common Term

Apply the distributive property by factoring out \(\sqrt{y}\) from each term. This gives us: \((7 + 1 + 2) \sqrt{y}\).

Key Concepts

Like Terms

In algebra, the term "like terms" refers to terms that have the same variables raised to the same powers. This means they can be easily combined together. In our exercise, the terms are \(7\sqrt{y}, \sqrt{y},\) and \(2\sqrt{y}\). Each term contains the square root of \(y\), making them like terms.When dealing with like terms:

• Ensure that the variables match exactly, including their exponents and roots.
• Only the coefficients (the numbers in front of the variables) are added or subtracted, while the variable part remains unchanged.

By identifying like terms, you simplify expressions by consolidating them into one term. In this case, since all terms are like terms, you can combine them into \(\left(7 + 1 + 2\right) \sqrt{y}\). This results in \(10\sqrt{y}\). This step simplifies further operations and makes the expression easier to handle.

Factoring Out

Factoring is a process where you identify and extract common factors from terms in an expression. This is crucial in simplifying expressions, particularly when using the distributive property, as seen in our example with radicals.Here’s how factoring out works:

• Identify common factors in terms, here it's \(\sqrt{y}\) for all terms.
• Factor out the common element, making it a single unit outside of a parenthesis.
• You’re left with a simplified expression inside the parenthesis, here’s how:
  • From \(7\sqrt{y} + \sqrt{y} + 2\sqrt{y}\) factor out \(\sqrt{y}\).
  • This becomes \((7 + 1 + 2)\sqrt{y}\).

Factoring out reduces complexity by focusing on the coefficients and brings clarity to solving equations involving multiple similar terms.

Radicals

Radicals, often dealing with roots, are expressions containing the square root, cube root, or other roots of numbers. They can initially look complex, but understanding their properties makes working with them manageable.Key things about radicals:

• The nth root of a number \(x\) is denoted as \(x^{1/n}\) or \(\sqrt[n]{x}\).
• A square root, the most common type, is represented by \(\sqrt{x}\).
• In our problem, \(\sqrt{y}\) is the radical part where \(y\) is the expression under the root.
• When combining radicals, they must be like radicals, meaning they have the same expression under the root.

To handle expressions with radicals effectively:

• Recognize when radicals are like terms so they can be combined.
• Be mindful of the properties of exponents which often help in simplifying radicals further.

By mastering radicals, you improve your skills to simplify and solve more complex algebraic expressions.