Problem 24 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{162 a^{4} b^{3}}+3-a b \sqrt{18 a^{2} b}-1 $$

Step-by-Step Solution

Answer

The simplest form is \(6 a^2 b \sqrt{2b} + 2\).

Step 1: Break Down the Radicals

First, simplify each radical individually. The expression is \( \sqrt{162 a^{4} b^{3}} + 3 - a b \sqrt{18 a^{2} b} - 1 \). Start with \( \sqrt{162 a^{4} b^{3}} \).

Step 2: Simplify \(\sqrt{162 a^{4} b^{3}}\)

Find the prime factorization of 162: \(162 = 2 \times 3^4\). Therefore, \(\sqrt{162} = \sqrt{2 \times 3^4} = 3^2 \sqrt{2} = 9 \sqrt{2}\).\Now simplify the variables: \(\sqrt{a^4} = a^2\) and \(\sqrt{b^3} = b \sqrt{b}\).Thus, \(\sqrt{162 a^4 b^3} = 9 a^2 b \sqrt{2b}\).

Step 3: Simplify \(a b \sqrt{18 a^2 b}\)

Similarly, factor 18: \(18 = 2 \times 3^2\). So, \(\sqrt{18 a^2 b} = \sqrt{2 \times 3^2 \times a^2 \times b} = 3a \sqrt{2b}\).Thus, the expression \(a b \sqrt{18 a^2 b}\) simplifies to \(3a^2 b \sqrt{2b}\).

Step 4: Substitute Simplified Radicals into Expression

Place the simplified radicals back into the original expression:\(9 a^2 b \sqrt{2b} + 3 - 3 a^2 b \sqrt{2b} - 1\).

Step 5: Combine Like Terms

Combine the terms that contain \(a^2 b \sqrt{2b}\) and the constants:\( (9 a^2 b \sqrt{2b} - 3 a^2 b \sqrt{2b}) + (3 - 1)\).

Step 6: Simplify Final Expression

Calculate each part:- \(9 a^2 b \sqrt{2b} - 3 a^2 b \sqrt{2b} = 6 a^2 b \sqrt{2b}\).- \(3 - 1 = 2\).Thus, the simplest form of the expression is \(6 a^2 b \sqrt{2b} + 2\).

Key Concepts

Radical Expressions

Radical expressions involve roots, like square roots or cube roots, of numbers or variables. They can look complex, but they are essential for many math problems. An important part of dealing with them is simplifying. This means making them as simple as possible without changing their value.

• To simplify a radical, find a way to express it using smaller numbers or variables.
• If the radicand (the number inside the radical) has perfect squares, these can be factored out of the radical.

For example, in the expression \(\sqrt{162 a^4 b^3}\), you can factor out the perfect squares to make it easier to understand. Once factored, you can rewrite the expression in a simpler form, such as \(9a^2b\sqrt{2b}\). This makes the math cleaner and calculations simpler.

Prime Factorization

Prime factorization is a tool that helps in simplifying radical expressions. It involves breaking down a number into its basic building blocks, which are prime numbers. **Why Use Prime Factorization?** Mainly to identify perfect squares within the radicand, making simplification easier.

For instance, the number 162 can be factorized into its prime factors as \(2 \times 3^4\). When simplifying \(\sqrt{162}\), recognizing \(3^4\) involves perfect squares allows you to simplify the expression to \(9\sqrt{2}\) since \(3^4 = (3^2)^2 = 9 \times 9\). Applying this principle to radical expressions helps in reducing the complexity of the expression considerably.

• Identify the prime factors of a number.
• Look for pairs of factors to help simplify the square root.

This technique applies to both numerical coefficients and variable parts within a radical.

Variable Expressions

Variable expressions contain variables along with numbers, which can make them appear complicated. Simplifying these expressions involves using properties of exponents and radicals. **The Basic Idea** is to factor the variables similarly to numbers and deal with them inside the radical.

For example, in the expression \(\sqrt{a^4}\), the property \(\sqrt{a^4} = a^2\) can be directly applied because every two of the same variable inside the radical come outside as one. The same method works for \(\sqrt{b^3}\), which simplifies to \(b\sqrt{b}\), taking one \(b\) out and leaving one inside the square root. This makes variable expressions straightforward when their factors are correctly identified.

Combining Like Terms

Combining like terms simplifies expressions by gathering and consolidating terms that have the same structure. **Why Combine Like Terms?** This helps to simplify the expression and see it in its simplest form.

• Like terms have identical variable parts.
• Combine their coefficients.

For example, in the expression \(9a^2b\sqrt{2b} - 3a^2b\sqrt{2b}\), we notice that both terms contain \(a^2b\sqrt{2b}\). By combining them, you subtract their coefficients: \(9 - 3 = 6\). The result is \(6a^2b\sqrt{2b}\).
Combining like terms benefits greatly in simplification, providing a cleaner and more manageable representation of expressions.