Problem 270 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\quad \sqrt{11}(-3+4 \sqrt{11})\) (b) \(\sqrt{3}(\sqrt{15}-\sqrt{18})\)

Step-by-Step Solution

Answer

For (a): -3 \sqrt{11} + 44. For (b): 3 \sqrt{5} - 3 \sqrt{6}.

Step 1: Distribute \( \sqrt{11} \) in Part (a)

Multiply \( \sqrt{11} \) with each term inside the parentheses. \[ \sqrt{11}(-3 + 4 \sqrt{11}) = \sqrt{11}\times(-3) + \sqrt{11}\times(4 \sqrt{11}) \]

Step 2: Simplify Each Term in Part (a)

Calculate the products: \[ \sqrt{11}\times(-3) = -3 \sqrt{11} \] and \[ \sqrt{11}\times(4 \sqrt{11}) = 4 \sqrt{11 \times 11} = 4 \sqrt{121} = 4 \times 11 = 44 \]

Step 3: Combine the Results in Part (a)

Add the simplified terms to get the final result: \[ -3 \sqrt{11} + 44 \]

Step 4: Distribute \sqrt{3} in Part (b)

Multiply \( \sqrt{3} \) with each term inside the parentheses: \[ \sqrt{3}(\sqrt{15} - \sqrt{18}) = \sqrt{3} \times \sqrt{15} - \sqrt{3} \times \sqrt{18} \]

Step 5: Simplify Each Term in Part (b)

Calculate the products: \[ \sqrt{3} \times \sqrt{15} = \sqrt{3 \times 15} = \sqrt{45} = 3 \sqrt{5} \] and \[ \sqrt{3} \times \sqrt{18} = \sqrt{3 \times 18} = \sqrt{54} = 3 \sqrt{6} \]

Step 6: Combine the Results in Part (b)

Combine the simplified terms to get the final result: \[ 3 \sqrt{5} - 3 \sqrt{6} \]

Key Concepts

radical expressions

Radical expressions involve roots such as square roots, cube roots, etc. They are defined by the symbol \(\frac{\text{}}{\underline{\phantom{xx}}}\). For example, \( \sqrt{11} \) is a radical expression where 11 is under the square root. Simplifying these expressions involves performing mathematical operations like multiplication or addition while keeping the rule that only like terms can be combined. For instance, you can't directly add \( \sqrt{2} \) and \( \sqrt{3} \) without simplifying further.

distributive property

The distributive property is a crucial algebraic principle used in simplifying expressions, including those with radicals. It allows you to multiply a single term by each term within a parenthesis. For example, in the exercise given:

• We have \( \sqrt{11} (-3 + 4 \sqrt{11}) \)
• We distribute \( \sqrt{11} \) to both terms inside the parenthesis, resulting in \( -3 \sqrt{11} + 4 \sqrt{11} \times \sqrt{11} \)

By distributing, we break down complex expressions into simpler components, making them easier to solve.

simplifying square roots

Simplifying square roots involves expressing them in the simplest form possible. This can involve breaking down the number under the root into its prime factors. When you encounter \( \sqrt{a \times b} \), it can be simplified to \( \sqrt{a} \times \sqrt{b} \). For example:

• In the solution, \( \sqrt{11} \times 4 \sqrt{11} \) is simplified to 44 by recognizing that \( \sqrt{11 \times 11} = \sqrt{121} = 11 \).
• For another example, \( \sqrt{3} \times \sqrt{15} = \sqrt{45} = 3 \sqrt{5} \), this follows the simplification and combination rules thoroughly.

The goal is to represent the numbers in a form that's easier to read and understand without changing their value.