Problem 16 - 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. $$ 3 x^{3} \sqrt{80}+2 \sqrt{125 x^{6}} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \( 22 x^3 \sqrt{5} \).

Step 1: Simplify the Radical

First, simplify the expression within the square roots. Start with \( \sqrt{80} \). We know \( 80 = 16 \times 5 \) and \( \sqrt{16} = 4 \), so \( \sqrt{80} = 4 \sqrt{5} \).

Step 2: Simplify the Variables

Consider the expression \( 3x^3 \sqrt{80} \). Using the result from Step 1: \( 3x^3 \sqrt{80} = 3x^3 \cdot 4\sqrt{5} = 12x^3 \sqrt{5} \).

Step 3: Simplify the Other Radical

Now simplify \( \sqrt{125 x^6} \). Notice \( 125 = 25 \times 5 \), so \( \sqrt{125} = 5\sqrt{5} \). For the variable part, \( \sqrt{x^6} = x^3 \). Thus, \( \sqrt{125 x^6} = 5x^3 \sqrt{5} \).

Step 4: Combine Like Terms

Combine the expressions \( 12x^3 \sqrt{5} + 2\sqrt{125x^6} \). From Step 3, \( 2 \sqrt{125x^6} = 2 \times 5x^3 \sqrt{5} = 10x^3 \sqrt{5} \). Now, add it to the term from Step 2: \( 12x^3 \sqrt{5} + 10x^3 \sqrt{5} = 22x^3 \sqrt{5} \).

Step 5: Final Answer

The expression \( 3 x^{3} \sqrt{80}+2 \sqrt{125 x^{6}} \) in its simplest form is \( 22 x^3 \sqrt{5} \).

Key Concepts

Understanding Radical Simplification

Simplifying radicals is about breaking down a square root expression into simpler components. A radical expression often houses numbers that can be factored among perfect squares, making it easier to simplify. For example, if you have \( \sqrt{80} \), you would identify that 80 can be broken down into \( 16 \times 5 \), where 16 is a perfect square. Simplifying \( \sqrt{16} \) gives us 4, so \( \sqrt{80} = 4\sqrt{5} \). By doing this, we turn a complex-looking expression into something more manageable.
Simplifying radicals using prime factorization or by identifying perfect squares allows you to perform computations more smoothly. It's like rearranging pieces of a puzzle until the picture becomes clear. Always look to simplify the radicand, the number inside the square root, by isolating its largest perfect square factor.

Combining Like Terms in Radical Expressions

When dealing with radical expressions, it is crucial to recognize like terms. Like terms in algebra are those expressions that have the same variables raised to the same powers. In the context of radicals and variable expressions, like terms also include having the same component under the radical.
For example, in an expression like \( 12x^3\sqrt{5} + 10x^3\sqrt{5} \), both terms contain \( x^3\sqrt{5} \). This commonality allows us to add them, much like combining simple numerical coefficients:

• Add the coefficients, 12 and 10, to get 22.
• Keep the radical and variable part unchanged: \(x^3\sqrt{5} \).

So, \( 12x^3\sqrt{5} + 10x^3\sqrt{5} \) combines neatly into \( 22x^3\sqrt{5} \). Recognizing and combining like terms helps simplify expressions significantly.

Demystifying Variable Expressions with Radicals

Variable expressions with radicals involve not just numbers but also variables inside or outside the square roots. A variety in which the operations and rules of simplification become a little more intricate.
For example, consider a term like \( \sqrt{125x^6} \). We simplify the number part to \( 5\sqrt{5} \), knowing that 125 breaks into 25, a perfect square, times 5. The variable part \( x^6 \) inside the square root can be seen as \( (x^3)^2 \), allowing us to pull \( x^3 \) out of the square root, just like any other perfect square.

• Apply the rule that \( \sqrt{x^n} = x^{n/2} \) for even \( n \).
• Thus, \( \sqrt{x^6} \) becomes \( x^3 \), simplifying the expression.

By applying consistent logic and rules to each part of the expression, even those involving multiple variables, simplification becomes much less intimidating.

Basics of Square Roots in Radical Expressions

The square root is a fundamental concept that underpins radical expressions. Understanding it deeply is key to mastering simplification. The square root of a number, written as \( \sqrt{a} \), is the number that, when multiplied by itself, gives \( a \).
Working with square roots often involves identifying any factors of the radicand (the number under the square root) that are perfect squares. Consider the square root \( \sqrt{80} \); by deconstructing 80 into 16 and 5, we observe that \( 16 \) is a perfect square, simplifying the expression to \( 4\sqrt{5} \).

• Always look for the largest perfect square factor to simplify efficiently.
• Rationalization often involves moving factors out from under the root.

Effectively handling radicals involves recognizing these patterns and applying simplification rules to transform expressions into their simplest form effortlessly, saving time and reducing potential errors.