Problem 286 - Elementary Algebra

Question

In the following exercises, simplify. $$ (7+2 \sqrt{m})(4+9 \sqrt{m}) $$

Step-by-Step Solution

Answer

28 + 71 \( \sqrt{m} \) + 18m

Step 1 - Expand the Expression Using the Distributive Property

First, use the distributive property (also known as the FOIL method for binomials) to expand the expression \((7+2 \, \sqrt{m})(4+9 \, \sqrt{m})\). Multiply each term in the first binomial by each term in the second binomial:\(7 \, (4 + 9 \, \sqrt{m}) + 2 \, \sqrt{m} \, (4 + 9 \, \sqrt{m})\).

Step 2 - Multiply Each Term

Multiply the terms:\(7 \, (4) + 7 \, (9 \, \sqrt{m}) + 2 \, \sqrt{m} \, (4) + 2 \, \sqrt{m} \, (9 \, \sqrt{m})\).This simplifies to:\(28 + 63 \, \sqrt{m} + 8 \, \sqrt{m} + 18m\).

Step 3 - Combine Like Terms

Combine the like terms in the expression:\(28 + 71 \, \sqrt{m} + 18m\).There are no further like terms to combine, so this is the simplified form.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It states that for any three numbers, say a, b, and c: \(a(b + c) = ab + ac\).When you see an expression like \((7+2 \sqrt{m})(4+9 \sqrt{m})\), you apply this property to expand it. Each term in the first binomial will multiply each term in the second binomial.

FOIL Method

The FOIL method is a specific application of the distributive property for binomials. FOIL stands for First, Outer, Inner, Last - representing the terms you need to multiply:

• First: Multiply the first terms in each binomial (7 * 4)
• Outer: Multiply the outer terms in the binomials (7 * 9\(\sqrt{m}\))
• Inner: Multiply the inner terms in the binomials (2\(\sqrt{m}\) * 4)
• Last: Multiply the last terms in each binomial (2\(\sqrt{m}\) * 9\(\sqrt{m}\))

So, for our example \((7+2\sqrt{m})(4+9\sqrt{m})\), we get these products: 28, 63\(\sqrt{m}\), 8\(\sqrt{m}\), and 18\(m\).

Combining Like Terms

In algebra, combining like terms simplifies an expression by adding or subtracting terms with the same variable components. In your expanded expression \(28 + 63\sqrt{m} + 8\sqrt{m} + 18m\), you can see that 63\(\sqrt{m}\) and 8\(\sqrt{m}\) are like terms because they have \(\sqrt{m}\). When combined, these terms add up to 71\(\sqrt{m}\).

Simplifying Expressions

Simplifying expressions involves performing all possible operations to transform an expression into its simplest form. Follow these steps:

• Apply the distributive property or FOIL method.
• Combine like terms whenever possible.

Using our example, after distributing and combining like terms, the final simplified expression becomes \(28 + 71\sqrt{m} + 18m\). Always look for any terms that can be further combined or simplified.