Problem 51 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ \sqrt{3 x}(\sqrt{3}-\sqrt{x}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 3\sqrt{x} - x\sqrt{3} \).

Step 1: Distribute the Square Root

We will distribute the term \( \sqrt{3x} \) across the terms inside the parentheses: \( (\sqrt{3} - \sqrt{x}) \). This is done by multiplying \( \sqrt{3x} \) with each term in the parentheses separately. The expression can be expanded as follows:\[\sqrt{3x} \cdot \sqrt{3} - \sqrt{3x} \cdot \sqrt{x}\]

Step 2: Simplify the First Term

We simplify the first product, \( \sqrt{3x} \cdot \sqrt{3} \). Since the property of square roots \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \, b} \) applies, combine the terms under a single square root:\[\sqrt{3x \cdot 3} = \sqrt{9x} = 3\sqrt{x}\]

Step 3: Simplify the Second Term

Simplify the second product, \( \sqrt{3x} \cdot \sqrt{x} \). Using the same property of square roots:\[\sqrt{3x \cdot x} = \sqrt{3x^2} = x\sqrt{3}\]This is because \( \sqrt{x^2} = x \).

Step 4: Combine the Simplified Terms

We now have two simplified terms: \( 3\sqrt{x} \) and \( x\sqrt{3} \). Put them together to get the final expression:\[3\sqrt{x} - x\sqrt{3}\]This is the simplified form of the original expression.

Key Concepts

Square Roots

The square root is a fundamental concept in mathematics. It represents a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. Square roots are symbolized by the radical sign \( \sqrt{ } \). You'll often encounter expressions where numbers or even variables appear under this symbol.
To clarify:

• \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \)
• \( \sqrt{x^2} = x \) when \( x \) is non-negative

Remember to always be cautious with square roots involving variables, as they can represent both positive and negative roots depending on the context.

Distribution of Square Roots

Distribution is a key principle in algebra, often applied in expressions to simplify complex terms. When dealing with square roots and distribution, you multiply the square root outside the parentheses by each term inside.
For example, consider an expression like \( \sqrt{a}(b - c) \). You would distribute as follows:

• Multiply \( \sqrt{a} \) by \( b \): \( \sqrt{a} \times b \)
• Multiply \( \sqrt{a} \) by \( c \): \( \sqrt{a} \times c \)
• The expression becomes: \( b\sqrt{a} - c\sqrt{a} \)

This process simplifies the original problem, making it easier to handle further operations.

Simplifying Radicals

Simplifying radicals means breaking down square roots to their simplest form. This involves utilizing properties of square roots, such as \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), to combine or separate terms.
In the expression \( \sqrt{3} \times \sqrt{3x} \), you use this property to combine them under one radical: \( \sqrt{9x} \). You can simplify further knowing that \( \sqrt{9} = 3 \), resulting in \( 3\sqrt{x} \).
Simplifying radicals not only tidies up your work but is also essential in solving more complicated algebraic equations.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their most concise form, making them easier to interpret or solve. This process combines like terms and simplifies any radicals or fractions involved. In context, the given expression \( 3\sqrt{x} - x\sqrt{3} \) is the result of algebraic simplification.
Simplification allows you to:

• Recognize equivalent expressions
• Solve equations more efficiently
• Make calculations more manageable

Understanding these principles aids in mastering more advanced concepts, helping to develop strong problem-solving skills in algebra.