Problem 64 - Intermediate Algebra

Question

Multiply. $$ 3 x^{1 / 2}(x+y) $$

Step-by-Step Solution

Answer

\(3x^{3/2} + 3x^{1/2}y\)

Step 1: Distribute the Multiplier

Apply the distributive property by multiplying the term outside the parenthesis, \(3x^{1/2}\), by each term inside the parentheses, \((x + y)\). You multiply \(3x^{1/2} \times x\) and \(3x^{1/2} \times y\).

Step 2: Simplify the First Term

Multiply the terms \(3x^{1/2} \times x\). This is equivalent to \(3x^{1/2 + 1}\). By adding the exponents, you get \(3x^{3/2}\).

Step 3: Simplify the Second Term

Multiply the terms \(3x^{1/2} \times y\) to get \(3x^{1/2}y\).

Step 4: Write the Final Expression

Combine the results from Steps 2 and 3 to obtain the final expression: \(3x^{3/2} + 3x^{1/2}y\). This is the expanded form of the original expression.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. In our exercise, we have a term with an exponent: \( x^{1/2} \). This is known as the square root, which is indeed equivalent to saying \( \sqrt{x} \). When dealing with exponents, the base is the number being multiplied, which in this case is \( x \). The exponent indicates how many times the base is used in the multiplication.
When we multiply terms with the same base, we add their exponents. For example, \( x^{a} \times x^{b} = x^{a+b} \). This property is key when simplifying the expression \( 3x^{1/2} \times x \). We add the exponents, \( 1/2 + 1 \), which gives us \( 3/2 \).
Understanding this basic rule of exponents helps simplify expressions and solve algebra problems more efficiently.

Algebraic Expressions Fundamentals

Algebraic expressions are a combination of numbers, variables, and operations. In our case, the expression is \( 3x^{1/2}(x + y) \).
An expression may include one or more variables, such as \( x \) or \( y \), and can also include constants (numerical coefficients), like the number 3 in our exercise.
Brackets or parentheses are used to group terms that you need to consider together when multiplying or applying other operations. Here, the distributive property comes into play by enabling us to multiply the term outside the parentheses, \( 3x^{1/2} \), by each term inside: \( x \) and \( y \).
Understanding the structure of algebraic expressions is essential in order to manipulate and simplify them correctly.

Simplifying Expressions with the Distributive Property

Simplifying expressions is a key skill in algebra, making complex expressions easier to handle. In the expression \( 3x^{1/2}(x + y) \), we apply the distributive property. This property states that \( a(b + c) = ab + ac \).
Here's how it works:

• Multiply \( 3x^{1/2} \times x \) first. Using rules of exponents, we find \( 3x^{3/2} \).
• Next, multiply \( 3x^{1/2} \times y \) to obtain \( 3x^{1/2}y \).

Once both terms are simplified, they are combined to form the expression: \( 3x^{3/2} + 3x^{1/2}y \).
The expression is now in its simplest expanded form, where each part is expressed as a sum of individual terms. This process highlights the power of the distributive property in breaking down and simplifying algebraic expressions.