Problem 13 - 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. $$ \sqrt{98 c^{5}}-\sqrt{18 c^{5}} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \( 4c^2\sqrt{2c} \).

Step 1: Simplify Each Radical

First, we need to simplify each radical expression independently. Begin with \( \sqrt{98c^5} \), which can be rewritten as \( \sqrt{49 \cdot 2 \cdot c^5} \). The number 49 is a perfect square (\(49 = 7^2\)), so we can take the square root, giving \( 7\sqrt{2c^5} \). Next, simplify \( \sqrt{18c^5} \), which is \( \sqrt{9 \cdot 2 \cdot c^5} \). The number 9 is a perfect square (\(9 = 3^2\)), so this becomes \( 3\sqrt{2c^5} \).

Step 2: Simplify Expression with the Same Base

Notice that both terms now have \( \sqrt{2c^5} \). Since both radicals are similar, we can subtract their coefficients like like terms. We have \( 7\sqrt{2c^5} - 3\sqrt{2c^5} \), which simplifies to \( (7 - 3)\sqrt{2c^5} = 4\sqrt{2c^5} \).

Step 3: Further Simplify the Variable Portion

Now simplify \( \sqrt{2c^5} \). Firstly, \( c^5 \) can be rewritten as \( c^4 \cdot c \) and since \( c^4 = (c^2)^2 \), it can be taken out from under the square root as \( c^2 \). Thus, \( \sqrt{2c^5} = \sqrt{2c^4c} = c^2 \sqrt{2c} \).

Step 4: Combine the Simplified Parts

Substitute the simplified version of the radical back into the expression. This gives us \( 4\cdot (c^2\sqrt{2c}) = 4c^2\sqrt{2c} \). This is the final simplified form of the original expression.

Key Concepts

Radicals in Algebraic Expressions

In algebra, a radical expression involves roots, represented by the radical sign \( \sqrt{} \). Radicals play a crucial role in simplifying non-integer powers, particularly when dealing with expressions that include roots such as square roots, cube roots, etc.

When simplifying radicals, the goal is to determine whether the expression within the radical sign has factors that are perfect powers, which makes it possible to extract integers from under the radical sign.

• An important step is to express the radicand (the number under the radical) as a product of its factors to identify perfect squares or other powers.
• In cases where you deal with variables, ensure that you account for even indices and non-negative assumptions if applicable.

Understanding radicals helps in mastering the simplification process, ensuring expressions are in their simplest forms before applying arithmetic operations.

Perfect Squares and Their Importance

Perfect squares are integers that can be expressed as the square of another integer. For example, 49 is a perfect square because it can be written as \(7^2\). Identifying these numbers is key in simplifying square root expressions.

In algebraic simplification, recognizing perfect squares allows you to simplify radicals effectively by taking the square root of the integer within the square root sign:

• When presented with an expression like \( \sqrt{98c^5} \), factorize to find the perfect square. Here, 98 can be broken down to \(49 \times 2 \), with 49 being \(7^2\).
• Knowing this, you can take the square root of 49, simplifying the expression to \(7\sqrt{2c^5}\).

Utilizing the concept of perfect squares allows one to efficiently simplify complex expressions, which is particularly useful when dealing with multiple terms.

Combining Like Terms

In algebra, "like terms" are terms that have the same variables raised to the same power. In the context of radicals, it's essential to identify and combine like terms to simplify expressions effectively.

• Take the expression \(7\sqrt{2c^5} - 3\sqrt{2c^5}\). Here, both terms share the same radical component: \(\sqrt{2c^5}\).
• These terms are "like terms" because they have identical radical parts. Therefore, you can subtract their coefficients, simplifying the expression to \((7-3)\sqrt{2c^5} = 4\sqrt{2c^5}\).

Simplifying expressions by combining like terms is important because it reduces complexity and brings clarity to mathematical problems. It’s a fundamental technique in algebra.

Variables in Radicals: Simplification Tips

When variables are involved in radicals, simplifying requires careful attention to the radicand's structure. In the expression \(\sqrt{2c^5}\), the goal is to maximize what can be expressed outside the radical sign.

• Firstly, express \(c^5\) as \(c^4 \cdot c = (c^2)^2 \cdot c\). Recognizing that \((c^2)^2\) is a perfect square, it can be removed from the radical as \(c^2\).
• This results in \(c^2\sqrt{2c}\), simplifying the variable component of the radical expression.

This step is crucial because separating variables from radicals can greatly simplify further algebraic manipulations and help in achieving the simplest form of the expression.