Problem 96 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (\sqrt{a b}-1)(\sqrt{a b}-2) $$

Step-by-Step Solution

Answer

The simplified expression is \(a b - 3 \sqrt{a b} + 2\).

Step 1: Use the distributive property

Start by applying the distributive property to multiply the binomials. This involves multiplying each term in the first binomial by each term in the second binomial: \((\sqrt{a b} - 1)(\sqrt{a b} - 2) = (\sqrt{a b})(\sqrt{a b}) + (\sqrt{a b})(-2) + (-1)(\sqrt{a b}) + (-1)(-2)\).

Step 2: Simplify each term

Now, simplify each term from the distribution:1. \((\sqrt{a b})(\sqrt{a b}) = a b\) because the square root of a number times itself is the number.2. \((\sqrt{a b})(-2) = -2 \sqrt{a b}\).3. \((-1)(\sqrt{a b}) = - \sqrt{a b}\).4. \((-1)(-2) = 2\).

Step 3: Combine Like Terms

After simplifying, combine like terms: \(a b - 2 \sqrt{a b} - \sqrt{a b} + 2\). The like terms are the terms with \(\sqrt{a b}\):\(-2 \sqrt{a b} - \sqrt{a b} = -3 \sqrt{a b}\). So, the expression becomes \(a b - 3 \sqrt{a b} + 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle which is essential when multiplying expressions, especially binomials\. This property states that when you multiply a single term by a sum of two or more terms, you distribute the multiplication over each term separately\. In simpler terms, for any numbers or expressions \( a, b, \) and \( c \), the distributive property can be expressed as:

• \( a(b + c) = ab + ac \)

In this example, we applied the distributive property to the binomials \((\sqrt{ab} - 1)(\sqrt{ab} - 2)\)\. By distributing each term in the first binomial across every term in the second, we methodically break down the multiplication process\. This step ensures that all terms are accounted for, leading to a fully expanded expression before any simplification takes place\.

Binomial Multiplication

Binomial multiplication is the process of multiplying two expressions that each contain two terms\. It's a common operation in algebra and requires careful distribution of each term\. When dealing with binomials, such as \((x - 1)(x - 2)\), each term in the first binomial must be multiplied by each term in the second binomial\. This is sometimes remembered with the acronym "FOIL," although the steps remain the same without it:\.

• First (multiply the first terms of each binomial)
• Outside (multiply the outer terms)
• Inside (multiply the inner terms)
• Last (multiply the last terms)

For our specific question, involving \((\sqrt{ab} - 1)(\sqrt{ab} - 2)\), each combination of terms is multiplied, resulting in four separate products which are then summed. Breaking it down this way ensures that all parts of the binomials are accounted for and makes the multiplication straightforward\.

Simplification

Simplification in algebra focuses on condensing an expression into its simplest form while maintaining its original value\. This involves combining like terms and reducing complex expressions\. In our example \((\sqrt{ab} - 1)(\sqrt{ab} - 2)\), we first expand using binomial multiplication and then simplify the terms:\.

• The expression \(\sqrt{a b} \cdot \sqrt{a b} \) simplifies to \(ab\), because the square root multiplied by itself results in the number under the radical\.
• We then have \( -2\sqrt{ab} - \sqrt{ab} \), which can be combined into \(-3\sqrt{ab}\).
• Finally, the constant sum of \(-1 \cdot -2\) simplifies to \(+2\)\.

This leads us to the final, simplified expression: \(ab - 3\sqrt{ab} + 2\)\. Simplification reduces the chance of errors in further mathematical operations and provides a clear, concise form of the result\.

Radicals

Radicals are mathematical symbols used to denote roots, and in particular, square roots are indicated by the radical symbol \(\sqrt{}\)\. For example, \(\sqrt{ab}\) denotes the square root of the product \(ab\)\. Comprehending radicals is crucial when handling expressions involving them, as they often appear in algebraic problems requiring simplification or multiplication\. When multiplying radicals such as \(\sqrt{ab} \cdot \sqrt{ab}\), it is helpful to remember that the product of two similar radicals equals the number under the root, i.e., \(ab\)\.

• This is because \((\sqrt{ab})^2 = ab\), reflecting how squaring and taking the square root are inverse operations\.

Understanding how to work with radicals expands one's ability to simplify complex expressions and solve equations efficiently\. This understanding is indispensable when progressing through more advanced mathematics topics where radicals often arise\.