Question

If \(a>0, b>0\) and \(a b=1\), then the e5510n \((1+a)(1+b)\) is: (b) 4 (a) 2 (d) \(1 / 4\) (c) 1

Short answer

Question: Given that \(a > 0\), \(b > 0\), and \(ab = 1\), find the value of \((1+a)(1+b)\). Answer: The simplified expression is \(a + b + 2\). However, none of the provided options match this expression, indicating an error in the problem statement or options.

Step-by-step solution

Understanding the given information

We are given that \(a > 0\), \(b > 0\), and \(ab = 1\). This means that both \(a\) and \(b\) are positive numbers and their product equals 1.

Expanding the expression

Now, we will expand the expression \((1+a)(1+b)\) using the distributive property. \((1+a)(1+b) = 1(1) + 1(b) + a(1) + a(b)\)

Using the given condition

Since we know that \(ab = 1\), we can replace \(a(b)\) in the expression with 1. So, we get \((1+a)(1+b) = 1 + b + a + 1\)

Simplifying the expression

The expression can be simplified as: \((1+a)(1+b) = 1 + a + b + 1 = a + b + 2\) Now, we can compare the simplified expression with the given options. (a) 2 (b) 4 (c) 1 (d) \(\frac{1}{4}\) None of the options match the simplified expression, so there might be an error in the problem statement or the given options.

Key concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are composed of variables and constants combined with arithmetic operations. For example, the expression \(1+a)(1+b)\) consists of two binomials that can be expanded into a more extended expression.

The process of 'expanding' systematically applies the basic arithmetic operations to simplify or rearrange the expression. This often involves using properties like the distributive property to combine like terms or to make the expression more workable for solving equations. In our exercise, we seek to expand and simplify the expression to find its value given certain conditions about the variables involved.

Understanding algebraic expressions is crucial as they represent mathematical relationships that can model real-world situations. Grasping this concept allows students to move onto more complex problem solving in algebra and beyond.

Positive Numbers

In mathematics, positive numbers are all the numbers greater than zero. They are on the right side of zero on the number line and represent quantities with magnitude. They are crucial in understanding the concept of magnitude and direction in various mathematical contexts.

In the given problem, we are told that both \(a\) and \(b\) are positive numbers, implying they are greater than zero. This information is not trivial; it sets the groundwork for determining the behavior of the algebraic expression, especially when it comes to the sign of the result after performing operations like addition, subtraction, multiplication, or division. For instance, the product of two positive numbers is always positive, which is why knowing \(a\) and \(b\) are positive leads us to conclude that \(ab=1\) is also a positive number.

Distributive Property

The distributive property is an essential rule in arithmetic that allows us to multiply a single term by each term within a parenthesis. It’s written as \(a(b + c) = ab + ac\). Applying this property makes it possible to expand algebraic expressions, like the one we encounter in our exercise \( (1+a)(1+b) \).

When we apply the distributive property to the expression, we multiply each term in the first binomial by each term in the second binomial. This step-by-step process ensures that all possible products are included in the expanded form.

The property not only simplifies expressions and makes them easier to work with, but also supports the understanding of other algebraic concepts such as factoring and polynomial multiplication. Correct application of this property is critical to arriving at the correct simplified form of an algebraic expression.