Problem 58 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 9+\sqrt{45}+\sqrt{20}+16 $$

Step-by-Step Solution

Answer

The simplified expression is \(25 + 5\sqrt{5}\).

Step 1: Identify Like Radicals

First, identify if there are any terms that are square root expressions. The terms involving square roots are \(\sqrt{45}\) and \(\sqrt{20}\).

Step 2: Simplify Square Roots

Simplify the square root expressions by finding perfect square factors.\[\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\]\[\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\]

Step 3: Combine Like Radicals

Combine the like radicals from the simplified expressions:\[3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}\]

Step 4: Combine All Terms

Now, add all constant terms and the combined radical:\[9 + 16 + 5\sqrt{5} = 25 + 5\sqrt{5}\]

Step 5: Write the Final Expression

The expression simplified by combining like radicals is:\[25 + 5\sqrt{5}\]

Key Concepts

Simplifying Square Roots

Simplifying square roots means rewriting the square root in its simplest form by breaking it down into more manageable parts. Imagine a square root as a puzzle that can be untangled and solved.To simplify, identify the largest perfect square factor of the number under the square root. A perfect square is a number that can be expressed as the square of an integer, like 4 (since 2x2=4) or 9 (since 3x3=9).

• Step 1: Look for a Perfect Square Factor

Breaking down the square root of 45 can be rewritten as \[\sqrt{45} = \sqrt{9 \times 5}\]. This is because 9 is the largest perfect square that divides 45.

• Step 2: Simplify Each Part

Now that we have \(\sqrt{9} \times \sqrt{5}\), we simplify \(\sqrt{9}\) to 3, as 9 is a perfect square. This leaves us with\[3\sqrt{5}\].Each specific step in simplifying brings the square root to a format that is easier to understand and combine later. Once you master this, solving problems involving square roots will become much easier!

Combining Like Terms

To solve algebraic expressions efficiently, combining like terms is a necessary skill. Like terms are terms in an expression that have identical variables and powers. In the context of this problem, it means combining square root expressions that share the same radical.Identifying and combining like radicals makes the expression simpler and clearer:

• Identify Like Radicals

In our exercise, \(\sqrt{45}\) simplifies to \(3\sqrt{5}\) and \(\sqrt{20}\) simplifies to \(2\sqrt{5}\). Once simplified, notice both expressions have the same radical, \(\sqrt{5}\).

• Combine the Coefficients

Combine these by simply adding their coefficients:\[3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}\].Combining like radicals is much like adding apples to apples; they must be the same before they can be added together. Once identified, you simply sum up their coefficients, making expressions much easier to work with.

Intermediate Algebra

Intermediate algebra sets the foundation for solving more complex algebraic equations. It involves a combination of simplifying expressions, manipulating variables, and understanding concepts such as like terms and radicals.

• Building Blocks

Mastering techniques like simplifying square roots and combining like terms fosters an understanding of how algebraic expressions can be manipulated. This ensures you are prepared for more advanced topics.

• Applying Knowledge

In our example, simplifying expressions through intermediate algebra involves multiple steps: - Identifying terms to be simplified - Breaking them down into smaller, more manageable parts - Combining terms smartly to simplify the entire expression Each step strengthens the ability to work with algebraic expressions logically. With practice, these concepts become second nature, making it easier to tackle equations of any complexity.