Problem 30 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{6}(7 \sqrt{3}+6) $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt{6}(7 \sqrt{3}+6)\) is \(21\sqrt{2}+6\sqrt{6}\)

Step 1: Apply Distributive Property

Apply the Distributive Property, which states that the product of a number and a sum, \(a(b+c)\), is equal to the sum of the products of the number and each term, \(ab+ac\). Hence, multiply each term inside the parenthesis by \(\sqrt{6}\).\n The expression thus becomes: \(\sqrt{6} \cdot 7\sqrt{3}+\sqrt{6} \cdot 6\)

Step 2: Simplify Square Root Terms

Use the rule \(\sqrt{a}\cdot \sqrt{b}=\sqrt{a \cdot b}\) to simplify the square root terms. This gives:\n\(7\sqrt{6 \cdot 3}+6\sqrt{6}\)\n Further simplification gives: \(7\sqrt{18}+6\sqrt{6}\)\n The square root of 18 can be simplified to \(\sqrt{9 \cdot 2}=3\sqrt{2}\), hence we have:\n\(7 \cdot 3\sqrt{2}+6\sqrt{6}\)

Step 3: Multiply Constants

Finally, multiply the constants to get the final simplified expression, which is:\n \(21\sqrt{2}+6\sqrt{6}\)

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic concept that makes handling expressions and equations manageable. It states that a single term outside a set of parentheses can be distributed to every term inside the parentheses. This is written mathematically as:

• \( a(b + c) = ab + ac \)

This property allows us to multiply and spread the term outside the parentheses across each term inside. Applying this property helps in expanding and simplifying algebraic expressions.
In the exercise, we start with \( \sqrt{6}(7\sqrt{3} + 6) \). Using the distributive property, we distribute \( \sqrt{6} \) to both the terms inside the parentheses, resulting in \( \sqrt{6} \cdot 7\sqrt{3} + \sqrt{6} \cdot 6 \).
This expands the expression, setting the stage for further simplification.

Square Root Simplification

Simplifying square roots makes expressions easier to work with. When you multiply square roots, you can consolidate them under one radical by using the multiplication rule:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)

This technique is particularly helpful when dealing with irrational numbers, such as square roots.
Continuing with the exercise, we have \( 7\sqrt{6 \cdot 3} + 6\sqrt{6} \). Notice that \( \sqrt{6 \cdot 3} \) can be further simplified. We compute: \( 6 \cdot 3 = 18 \), giving us \( \sqrt{18} \).
The square root of 18 can be simplified by breaking it down: \( \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \). So, our expression becomes \( 7 \cdot 3\sqrt{2} + 6\sqrt{6} \).
This process reduces the complexity of radicals, helping achieve a simpler form of the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can be constants, variables, coefficients, and include operations like addition, subtraction, multiplication, and division.
Understanding how to manipulate these expressions is key to solving algebra problems. Key operations include:

• Combining like terms
• Using the distributive property
• Simplifying radicals and expressions

In the exercise, the given algebraic expression is \( \sqrt{6}(7\sqrt{3} + 6) \). By applying the distributive property and simplifying the radicals step-by-step, we reach the final simplified expression: \( 21\sqrt{2} + 6\sqrt{6} \).
This involves each core concept like multiplication, combining radicals, and simplifying results, which embody the importance of algebraic manipulation skills.