Problem 32 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 3 \sqrt{7} t(2 \sqrt{7 t}+3 \sqrt{3 t^{2}}) $$

Step-by-Step Solution

Answer

The simplified expression is \(42t \sqrt{t} + 9t^2 \sqrt{21}.\)

Step 1: Distribute the First Term

Begin by distributing the first term \(3 \sqrt{7} t\) to each term inside the parentheses. We'll handle the expression \(2 \sqrt{7t}\) first. The expression becomes \((3 \sqrt{7} t) \cdot (2 \sqrt{7t})\).

Step 2: Multiply and Simplify the First Distribution

Multiply the coefficients: \(3 \times 2 = 6\). For the square roots, \( \sqrt{7} \times \sqrt{7t} = \sqrt{49t} = 7 \sqrt{t}\). Combine these to get: \[6 \times 7 \sqrt{t} \times t = 42t \sqrt{t}.\]

Step 3: Distribute the Second Term

Next, distribute \(3 \sqrt{7} t\) to the second term inside the parentheses: \((3 \sqrt{7} t) \cdot (3 \sqrt{3t^2})\).

Step 4: Multiply and Simplify the Second Distribution

Multiply the coefficients: \(3 \times 3 = 9\). For the square roots, \( \sqrt{7} \times \sqrt{3t^2} = \sqrt{21t^2} = t\sqrt{21}\) since \(\sqrt{t^2} = t\). Combine these to get: \[9 \times t \times t \sqrt{21} = 9t^2 \sqrt{21}.\]

Step 5: Combine Like Terms

Combine the results from Step 2 and Step 4 to form the single expression: \(42t \sqrt{t} + 9t^2 \sqrt{21}.\) This expression is the simplified result of the multiplication.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term across terms inside parentheses. In the original exercise, we use the distributive property to simplify the expression:

• Identify the term outside the parentheses, which is \(3 \sqrt{7} t\), and multiply it with each term inside the parentheses individually.
• First, multiply with \(2 \sqrt{7t}\), and then separately with \(3 \sqrt{3 t^{2}}\).

By carefully executing the distributive property, we can handle the multiplication one step at a time, ensuring a clear simplification process. This method is essential when working with more complex algebraic expressions and is especially useful with radicals because it organizes the multiplication process neatly.

Multiplying Radicals

When multiplying radicals, it's important to handle both the coefficients and the radicands (the numbers inside the square root) separately. In the exercise, we encounter expressions like \(3 \sqrt{7} \times 2 \sqrt{7t}\). Here's how to approach this:

• First, multiply the coefficients: \(3 \times 2 = 6\).
• Then, multiply the radicals: \(\sqrt{7} \times \sqrt{7t} = \sqrt{49t}\).
• Simplify if possible: since \(\sqrt{49t} = 7\sqrt{t}\), combine it with the coefficient: \(6 \times 7\sqrt{t} = 42t\sqrt{t}\).

This step-by-step approach makes handling radicals straightforward. By separately addressing coefficients and radicals, we simplify the expression effectively, allowing clearer paths to further simplify the entire algebraic expression.

Breaking Down Algebraic Expressions

Algebraic expressions involving variables and radicals can sometimes look complex, but breaking them down helps in simplifying. In our example, we dealt with an expression like \(42t \sqrt{t} + 9t^2 \sqrt{21}\). Here's how to understand them better:

• Each part of the expression consists of a mix of coefficients, variables, and radicals organized based on multiplication rules.
• The expression \(42t \sqrt{t}\) combines a coefficient, a variable \(t\), and a radical \(\sqrt{t}\).
• Similarly, \(9t^2 \sqrt{21}\) has a coefficient (9), a squared variable \(t^2\), and the radical \(\sqrt{21}\).

By understanding how each part of the expression contributes to the overall form, it becomes easier to manipulate and simplify, an essential skill in algebra. This breakdown facilitates seeing how terms can be combined, organized, or further simplified if needed.