Problem 30 - Elementary Algebra Within Reach

Question

In Exercises 19-36, expand the expression as a product of factors. $$ 2(x z)^{4} $$

Step-by-Step Solution

Answer

The expansion of \(2(x z)^{4}\) as a product of factors is \(2x^4 . z^4\).

Step 1: Distribute the Exponent to Each Term Within the Parentheses

Recall that \((a.b)^n = a^n . b^n\). So, using this rule, we have \(2(x.z)^4 = 2(x^4 . z^4)\).

Step 2: Apply Multiplication

2 multiplied by \(x^4\) and \(z^4\) gives us \(2x^4 . z^4\) as the final expression.

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and solve equations efficiently. It is often represented by the formula: \( a(b + c) = ab + ac \) where \( a \) can be distributed over the addition inside the parentheses to create two terms, \( ab \) and \( ac \).

In the context of our exercise, the distributive property is slightly modified when dealing with exponents. Here, rather than distributing a coefficient over a sum, we distribute the exponent over the terms within the parentheses: \( (xz)^4 \) becomes \( x^4z^4 \) after distributing the power of 4 to both \( x \) and \( z \) individually. This step is crucial as it sets the stage for the multiplication that follows.

Exponents in Algebra

Understanding exponents, or powers, in algebra is essential for expanding expressions correctly. An exponent indicates how many times a number, known as the base, is multiplied by itself. The rule \( (ab)^n = a^n b^n \) used in our exercise is a powerful exponent rule that simplifies the process of distributing powers over products.

Visualizing Exponents

Imagine having four \( xz \) pairs and multiplying them all together: \( (xz)(xz)(xz)(xz) \). With algebraic shorthand, we write this as \( (xz)^4 \). The exponent rule then allows us to express this as \( x^4z^4 \) without writing out all four pairs. It is one of many exponent laws that make algebra both more efficient and easier to grasp.

In practice, correctly applying exponent rules ensures accuracy when dealing with more complex expressions involving powers.

Multiplication of Polynomials

When expanding expressions, we often encounter the multiplication of polynomials, which involves distributing terms and combining like terms. In our exercise, however, we're looking at a simplified scenario where we only have monomials—single-term expressions.

Once we have distributed the exponent to get \( x^4 \) and \( z^4 \) from \( (xz)^4 \) in our previous steps, we proceed to multiply these monomials by the coefficient 2. We apply the basic multiplication rule of combining the coefficient with the monomials to get the final expanded form, \( 2x^4z^4 \).

In a broader sense, when multiplying polynomials with multiple terms, we would use the FOIL method (First, Outer, Inner, Last) or a similar strategy to expand and simplify the expression. Here, with just one term each, the process is straightforward.