Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ 2 \sqrt{a b} \cdot 2 \sqrt{a b^{2}} $$

Step-by-Step Solution

Answer

The simplest form is \( 4ab \sqrt{b} \).

Step 1: Apply the Multiplication Property of Square Roots

Recall that the multiplication property of square roots states that if you have \( \sqrt{x} \cdot \sqrt{y} \), it is equal to \( \sqrt{x \cdot y} \). Thus, we can combine the two square root expressions: \( \sqrt{a b} \cdot \sqrt{a b^{2}} = \sqrt{(a b) \cdot (a b^{2})} \).

Step 2: Multiply the Radicands

Multiply the terms inside the square roots. We have \( (a b) \cdot (a b^{2}) = a^{2} b^{3} \). So, our expression now is \( \sqrt{a^{2} b^{3}} \).

Step 3: Simplify the Radicals

Simplify the square root of the expression. Since \(a^2\) is a perfect square, its square root is \(a\). Similarly, \(b^3 = b^2 \cdot b\), so the square root of \(b^2\) is \(b\), and \(b\) remains in the square root: \( \sqrt{a^{2} b^{3}} = a b \sqrt{b} \).

Step 4: Distribute the Coefficient

From the original expression, we had coefficients \(2 \times 2 = 4\). Multiply this coefficient by the simplified square root expression: \(4 \cdot a b \sqrt{b} = 4ab \sqrt{b}\).

Key Concepts

Simplifying Square Roots

Square roots might seem tricky at first, but simplifying them is all about making the radical expression easier to understand. The primary goal is to identify and remove all perfect square factors from underneath the square root sign. When you simplify a square root, you're essentially trying to "break down" the number or expression inside the square root into an easier form.
To simplify a square root expression, follow these steps:

• Identify any perfect square factors within the radicand (which is the term under the square root). A perfect square is a number that can be expressed as the square of an integer, for example, 4, 9, 16, and so on.
• For any perfect square factor, take its square root and move it outside the radical sign.
• Leave any non-perfect square factors under the square root.

By doing these steps, you simplify the expression, making it less complex and more manageable for further operations.

Perfect Squares

Understanding perfect squares is crucial when working with square roots, as they allow us to simplify radical expressions efficiently. A perfect square is a number obtained by squaring an integer. For instance, if you take the number 3 and square it, you get 9, which is a perfect square.
Here are some characteristics of perfect squares that are helpful in math:

• They always have an odd number of total factors. For example, 36 has nine factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36. The presence of a middle factor, in this case, 6 (since 6×6 = 36).
• All perfect squares are non-negative. This means they are either zero or positive numbers.
• Recognizing perfect squares helps in both factoring and simplifying expressions, making calculations less prone to error.

In the given example, recognizing that \(a^2\) and \(b^2\) are perfect squares allows us to simplify \(\sqrt{a^2b^3}\) to \(ab\sqrt{b}\).

Multiplication Property of Square Roots

The multiplication property of square roots is an essential tool simplifying expressions involving radicals. This rule asserts that the product of square roots is equal to the square root of a product. In mathematical terms, \(\sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y}\).
Here's how you can apply this property:

• This property allows you to combine radical terms into a single square root, simplifying calculations involving multiple root terms.
• Applying this property correctly can often make an otherwise complex expression much simpler, as it combines the terms and reduces clutter.
• In our example, this property helps us to simplify \(\sqrt{ab} \cdot \sqrt{ab^2}\) into \(\sqrt{a^2b^3}\), setting the stage for further simplification.

Understanding and effectively using the multiplication property of square roots are vital skills that enable you to manage and work with expressions involving radicals effectively.