Problem 1 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{3} \cdot \sqrt{3} $$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Understand the Expression

We are given the expression \( \sqrt{3} \cdot \sqrt{3} \). Our task is to simplify it. This involves using properties of square roots and multiplication.

Step 2: Apply the Multiplication Property of Square Roots

Recall that the property of square roots \( \sqrt{a} \cdot \sqrt{a} = a \) can be applied when both square roots are of the same number. Since both numbers under the square root are the same, i.e., 3, we can rewrite the expression as \( \sqrt{3 \times 3} \).

Step 3: Simplify the Expression

Now we simplify \( \sqrt{3 \times 3} \) to \( \sqrt{9} \), because 3 multiplied by 3 equals 9.

Step 4: Calculate the Square Root of 9

The next step is to find the square root of 9. Since \( 3 \times 3 = 9 \), the square root of 9 is 3. Thus, \( \sqrt{9} = 3 \).

Step 5: Conclusion

Hence, the expression \( \sqrt{3} \cdot \sqrt{3} \) simplifies to 3.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics and are denoted by the radical symbol \( \sqrt{} \). Understanding square roots means knowing that they represent a value which, when multiplied by itself, yields the original number. For example, \( \sqrt{9} = 3 \) because 3 times 3 equals 9. If you have a square root of a number, you're essentially looking for a number that can be squared to reach that original number. This concept is crucial when simplifying expressions, especially when dealing with like terms under the radical.

• A square root is simply one of two equal factors of a given number.
• For example, both \( 2 \) and \( -2 \) are square roots of \( 4 \), but in simplified form we always take the non-negative one.
• In expressions, it's important to carefully match numbers under the square root before applying mathematical operations.

Though the definition is simple, the application becomes a bit more technical when square roots appear in more complex equations or need to be simplified with operations like multiplication.

Multiplication Property

The multiplication property of square roots allows us to multiply two square roots together: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \). This property is incredibly useful when simplifying expressions or solving equations involving radicals. It means you can take each radical's value, multiply them together under a single square root, and simplify from there. For example, in the expression \( \sqrt{3} \cdot \sqrt{3} \), using this property, we combine them under one square root symbol: \( \sqrt{3 \times 3} \). This product is \( \sqrt{9} \), which simplifies further to 3.

• This property only applies when you're multiplying radicals; it doesn't work the same way for addition and subtraction.
• It simplifies dealing with large numbers by allowing you to "merge" the square roots into a single expression.
• Always be careful that the numbers you're working with fit under the square root across the equation.

This property assures simplified calculations, especially with positive variables since their multiplication doesn't impact the ability to compute square roots.

Positive Variables

When working with expressions involving square roots and multiplication, assuming positive variables is a standard practice. Positive variables simplify matters significantly across computations because they avoid complex numbers or negative results under the square root.Positive variables ensure our results remain in the realm of real numbers, facilitating easier simplification and clearer interpretation of results. For instance, in \( \sqrt{3} \cdot \sqrt{3} \), knowing that "3" is positive means we immediately assume the resultant manipulations will not involve any imaginary components.

• Positive variables prevent encountering situations (like \( \sqrt{-1} \)) that require new tools like imaginary numbers.
• They make it straightforward to apply mathematical properties since you're not worrying about the additional complexities introduced by negative or zero values.
• This clarity is why many exercises, like the one we've discussed, start by stating that all variables are positive.

In summary, working with positive values keeps mathematical operations clean and accessible, making simplifying expressions a more intuitive process.