Problem 67 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{5 x}-3 \sqrt{2})(\sqrt{5} x-3 \sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified form of the product is \( 5x\sqrt{x} - 3\sqrt{15x} - 3x\sqrt{10} + 9\sqrt{6} \).

Step 1: Expand the Expression

Use the distributive property (also known as FOIL for binomials) to expand \[ (\sqrt{5x} - 3\sqrt{2})(\sqrt{5}x - 3\sqrt{3}) \].This gives us four terms: 1) \( \sqrt{5x} \cdot \sqrt{5}x = (\sqrt{5} \cdot \sqrt{5}) \cdot (x \cdot \sqrt{x}) = 5x\sqrt{x} \),2) \( \sqrt{5x} \cdot -3\sqrt{3} = -3\sqrt{15x} \),3) \(-3\sqrt{2} \cdot \sqrt{5}x = -3x\sqrt{10} \),4) \(-3\sqrt{2} \cdot -3\sqrt{3} = 9\sqrt{6} \).

Step 2: Combine Like Terms

Combine the terms obtained from the expansion.The expanded expression is:\[ 5x\sqrt{x} - 3\sqrt{15x} - 3x\sqrt{10} + 9\sqrt{6} \].Note that there are no like terms to combine further since all terms involve different radicals. So this is the expression in its simplest form.

Key Concepts

Distributive Property

To start with, the distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term in a binomial or polynomial. This is often remembered as the FOIL method when applied to binomials. The acronym FOIL stands for First, Outside, Inside, and Last, referring to the order in which we multiply the terms from one binomial with those in another.

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

When using the distributive property in the example problem, we take care to apply this pattern carefully over the radicals and ensure proper handling of signs in the multiplications.

Binomials

A binomial is simply an algebraic expression with two terms usually connected by a plus or minus sign. In the context of the given exercise, our binomials are \(\sqrt{5x} - 3\sqrt{2} \) and \(\sqrt{5}x - 3\sqrt{3} \).

Each part of the binomial forms an important aspect during multiplication:

• The term \(\sqrt{5x}\) in the first binomial is paired with two terms in the second binomial to produce two unique products with roots.
• The constant \(- 3\sqrt{2}\) in the first binomial interacts with both terms in the second binomial, yielding products of constants with roots, handled similarly by separations of numerical and radical parts.

When working with binomials involving radicals, one must keep close attention to the operations done on both coefficients and radicands.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to understand or work with. When dealing with algebraic expressions involving radicals and both coefficients and variables, each term is simplified by examining its components.

In the example, after using the distributive property, we expand to get terms such as \(5x\sqrt{x}, -3\sqrt{15x}, -3x\sqrt{10},\) and \(9\sqrt{6} \).

• Each term needs to be checked for possible simplification, often by looking at the factors within the radicals for possible squares.
• In cases where the expressions involve complex radicals or variable parts, each term must remain separate if no other terms share identical radicands and coefficients outside the square roots.

The final step in the problem shows us that sometimes once expanded, no further simplification is possible, as there are no like terms to combine under these conditions.