Question

\(3\left(11^{1 / 4}\right)+9\left(11^{1 / 4}\right)\)

Short answer

The simplified form of the expression \(3\left(11^{1 / 4}\right)+9\left(11^{1 / 4}\right)\) is \(12(11^{1 / 4})\).

Step-by-step solution

Identify Common Factor

Identify the common factor in both terms of the expression. In this case, the common factor is \(11^{1 / 4}\), which is present in both \(3(11^{1 / 4})\) and \(9(11^{1 / 4})\).

Apply the Distributive Property in Reverse

Use the distributive property in reverse (also referred to as factoring), this means it will be changed from the format \(ab + ac\) to the format \(a(b + c)\). In the given exercise, this means it will be factored from \(3(11^{1 / 4}) + 9(11^{1 / 4})\) to \((3 + 9)(11^{1 / 4})\).

Simplify the Expression

Simplify the expression obtained in step 2. This yields \((3 + 9)(11^{1 / 4}) = 12(11^{1 / 4})\).

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that relates both multiplication and addition. It states that the product of a number and a sum is equal to the sum of the individual products of the number and each addend. In terms of a mathematical expression, the distributive property can be written as \( a(b + c) = ab + ac \). This property is useful because it allows for the multiplication to be distributed across the terms inside the parentheses.

In the exercise we've looked at, the distributive property is used in reverse. This means instead of expanding the expression, we condense it into a factored form. We are using the reverse of \( ab + ac = a(b + c) \) to write the expression \(3(11^{1/4}) + 9(11^{1/4})\) as \((3 + 9)(11^{1/4})\).

• This reverse usage is critical for simplifying expressions, as it helps in combining like terms and finding common factors.
• Understanding how to apply the distributive property both ways is crucial for manipulating algebraic expressions effectively.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form. This means it has the fewest possible terms or factors and no like terms remaining. Simplifying is often achieved by:

• Combining like terms.
• Using mathematical properties, like the distributive property.
• Eliminating parentheses by applying multiplication.

In the given problem, after using the distributive property in reverse, we simplify the expression from \((3 + 9)(11^{1/4})\) to \(12(11^{1/4})\).

This simplification is achieved by calculating the sum of \(3 + 9\), which gives \(12\). Then the expression becomes \(12(11^{1/4})\), which is the simplest form of the original expression. Simplifying is essential because it gives cleaner, more manageable terms that are easier to work with in further mathematical operations.

Common Factors

Finding common factors in expressions is a key skill in algebra. A common factor is a number or algebraic term that divides exactly into each term of the expression, without any remainders. Identifying these can significantly aid in simplifying algebraic expressions.

In our problem, the common factor is \(11^{1/4}\), present in both terms \(3(11^{1/4})\) and \(9(11^{1/4})\). Recognizing this common factor is crucial because it allows us to use the distributive property in reverse effectively, thereby factoring the expression to \((3 + 9)(11^{1/4})\).

• When you factor out the common factor, you simplify complex expressions and make calculations easier.
• This technique of finding and using common factors is used extensively in solving algebra problems, especially those involving polynomials and equations.

Working with common factors is like finding a key piece in a puzzle that helps unravel the entire expression.