Problem 252 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(4 \sqrt{6 t^{2}}\right)\left(3 \sqrt{3 t^{2}}\right) $$

Step-by-Step Solution

Answer

36 t^{2} \sqrt{2}

Step 1: Break Down the Expression

First, identify the separate components of the expression. The given expression is \ \( \left(4 \sqrt{6 t^{2}}\right) \left(3 \sqrt{3 t^{2}}\right) \).

Step 2: Multiply the Coefficients

Next, multiply the coefficients (numbers outside the square roots). \ \( 4 \times 3 = 12 \). So the expression becomes \ \( 12 (\sqrt{6 t^{2}} \cdot \sqrt{3 t^{2}}) \).

Step 3: Multiply the Radicals

Multiply the contents inside the square roots. \ \( \sqrt{6 t^{2}} \cdot \sqrt{3 t^{2}} = \sqrt{(6 t^{2}) (3 t^{2})} = \sqrt{18 t^{4}} \).

Step 4: Simplify the Radical Inside the Square Root

Simplify the expression inside the square root. \ \( 18 t^{4} \) can be factored as \ \( 18 = 9 \cdot 2 \) and \ \( t^{4} = (t^{2})^{2} \), so it becomes \ \( \sqrt{9 \cdot 2 \cdot (t^{2})^{2}} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{(t^{2})^{2}} = 3 \sqrt{2} \cdot t^{2} \).

Step 5: Combine the Results

Finally, combine the coefficient from Step 2 with the simplified radical from Step 4. \ \( 12 \cdot 3 \sqrt{2} \cdot t^{2} = 36 t^{2} \sqrt{2} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots. In our exercise, we work specifically with square roots. A square root expression looks like this: \( \sqrt{a} \). It means we are looking for a number which, when multiplied by itself, gives \( a \). For example, \( \sqrt{9} \) is 3 because 3 multiplied by 3 equals 9. In algebra, we often encounter more complex expressions like \( \sqrt{6 t^{2}} \). This includes variables and coefficients inside the radical.

Coefficients

Coefficients are the numerical part of an algebraic term. In our exercise, we have expressions like \( 4 \sqrt{6 t^{2}} \) where 4 is the coefficient. When simplifying expressions, it's important to handle coefficients properly. In our example, we first multiply the coefficients separately from the radicals. Here, we multiplied 4 and 3 to get 12. This step helps to simplify the expression by dealing with the numbers outside the square roots first.

Multiplying Radicals

Multiplying radicals like \( \sqrt{6 t^{2}} \) and \( \sqrt{3 t^{2}} \) involves combining the numbers under the square roots first. Since \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), we can multiply the terms inside the radicals together. In this case, \( 6 t^{2} \cdot 3 t^{2} \) gives us \( \sqrt{18 t^{4}} \). Multiplying the contents inside the radicals before taking the square root simplifies the process.

Simplifying Square Roots

To simplify square roots, factor the number to find perfect squares. For example, \( 18 = 9 \cdot 2 \). Since \( 9 \) is a perfect square of 3, we rewrite \( \sqrt{18} \) as \( 3 \sqrt{2} \). Similarly, \( \sqrt{(t^{2})^{2}} = t^{2} \). Combining these, we simplify \( \sqrt{18 t^{4}} \) to \( 3 t^{2} \sqrt{2} \). Finally, multiply this result by the original coefficient to get \( 36 t^{2} \sqrt{2} \), which is our simplified expression.