Problem 121 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt{x(x+3)} \sqrt{x^{3}(x+3)} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^2 (x+3) \).

Step 1: Multiply the Radicals

To multiply radical expressions, multiply the expressions under the radicals: \[ \sqrt{x(x+3)} \times \sqrt{x^3(x+3)} = \sqrt{[x(x+3)] \cdot [x^3(x+3)]} \] Combine the expressions under a single square root.

Step 2: Simplify the Expression Inside the Radical

Distribute and organize the terms inside the radical: \[ [x(x+3)] \cdot [x^3(x+3)] = x \cdot (x+3) \cdot x^3 \cdot (x+3) \] = \[ x^1 \cdot x^3 \cdot (x+3)^2 = x^4 \cdot (x+3)^2 \] This step shows the expressions combined and simplified with like terms.

Step 3: Apply Radical Rules for Simplification

Now that we have \( \sqrt{x^4 (x+3)^2} \), we can apply the rule that \( \sqrt{a^2} = a \) when \( a \geq 0 \). \[ \sqrt{x^4 (x+3)^2} = \sqrt{x^4} \cdot \sqrt{(x+3)^2} \] \[ = x^2 \cdot (x+3) \] Since \( x \) and \( x+3 \) are non-negative, we do not have to worry about absolute values.

Key Concepts

Addition of Radical Expressions

Adding radical expressions is much like adding numbers. However, radicals need to be alike or have the same index and radicand (the number or expression inside the root) to be combined. For example, to add \( \sqrt{5} + \sqrt{5} \), you would combine them as \( 2\sqrt{5} \). If they are not the same, like \( \sqrt{3} + \sqrt{5} \), you cannot add them together. Always simplify the radical expressions as much as possible before adding. This can sometimes lead to finding like radicals that weren't initially obvious.

Subtraction of Radical Expressions

Just like with addition, subtracting radical expressions also requires the radicands to be the same. When you subtract \( 3\sqrt{6} - 2\sqrt{6} \), you simply do \( (3-2)\sqrt{6} = \sqrt{6} \). If the radicands are different, such as \( \sqrt{7} - \sqrt{3} \), then they cannot be subtracted in a simplified form. Always check for possible simplification of the radicals first. This might involve factoring or other earlier algebraic simplifications that can make subtraction possible.

Multiplication of Radical Expressions

Multiplying radical expressions is straightforward if you follow the rule \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). In the provided exercise, expressions like \( \sqrt{x(x+3)} \text{ and } \sqrt{x^3(x+3)} \) were multiplied. This is done by multiplying the radicands: \( \sqrt{[x(x+3)] \cdot [x^3(x+3)]} \).

• Simplify by distributing terms: \([x(x+3)] \cdot [x^3(x+3)] = x^4(x+3)^2\).
• Combine terms, and then costradicalize if possible, collapsing back into a simpler expression.

After simplifying inside the square root as in the steps, the square root is broken down into separate roots, applying basic square root properties like \( \sqrt{x^4} = x^2 \).

Rationalizing the Denominator

Rationalizing the denominator helps in eliminating any radical from the denominator of a fraction for clarity and conventional results. If you encounter a fraction like \( \frac{a}{\sqrt{b}} \), multiply both the numerator and the denominator by \( \sqrt{b} \) to make the denominator a rational number:

• This can be expressed as \( \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a \sqrt{b}}{b} \).
• After multiplication, adjust the expression for any remaining squares to further simplify.

Rationalizing simplifies expressions, making it easier to add, subtract, and compare fractions, as well as fit conventional solution formats for proper math communication.