Problem 31 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. $$ (z+w)(z+w)(z+w)(z-w)(z-w) $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $(z+w)(z+w)(z+w)(z-w)(z-w)$ in exponential notation. Answer: $z^5 + w^5 - z^4w + 3z^4w - 6z^3w^2 + z^3w^2 - 6z^2w^3 + 3z^2w^3 - 2z^2w^3$

Step 1: Expand the expression

To expand the expression, we will use the distributive property (multiplying each term in one set of parentheses with each term in another set of parentheses). We have the expression: $$ (z+w)(z+w)(z+w)(z-w)(z-w) $$ First, expand the products of the first three pairs of parentheses: $$ ((z+w)(z+w)(z+w))(z-w)(z-w) $$ $$ (z^3+3z^2w+3zw^2+w^3)(z-w)(z-w) $$

Step 2: Continue expanding

Next, expand the product of the last expression and the fourth pair of parentheses: $$ (z^3+3z^2w+3zw^2+w^3)(z-w)^2 $$

Step 3: Expand the last square term

Expand the square term of (z-w) before multiplying with the expression: $$ (z^3+3z^2w+3zw^2+w^3)(z^2-2zw+w^2) $$

Step 4: Multiply the expanded expressions

Now, we multiply the expressions by using the distributive property again: $$ z^5 - 2z^4w + z^3w^2 + 3z^4w - 6z^3w^2 + 3z^2w^3 + 3z^3w^2 - 6z^2w^3 + 3zw^4 + z^3w^2 - 2z^2w^3 + w^5 $$

Step 5: Combine like terms

Finally, combine the terms with the same power to get the expression in exponential notation: $$ z^5 + z^3w^2 - z^3w^2 - 2z^4w + 3z^4w + 3z^3w^2 - 6z^3w^2 + z^3w^2 - 6z^2w^3 + 3z^2w^3 - 2z^2w^3 + w^5 $$ $$ z^5 + w^5 +(z^3w^2-z^3w^2) - z^4w + 3z^4w - 6z^3w^2 + z^3w^2 - 6z^2w^3 + 3z^2w^3 - 2z^2w^3 $$ The final expression in exponential notation is: $$ z^5 + w^5 - z^4w + 3z^4w - 6z^3w^2 + z^3w^2 - 6z^2w^3 + 3z^2w^3 - 2z^2w^3 $$

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). Think of it as a way to represent numbers and their relationships using symbols. For example, the algebraic expression for the perimeter of a rectangle with length 'l' and width 'w' would be expressed as \(2l + 2w\).

When working with algebraic expressions, we often need to manipulate them to simplify or solve equations. This could involve a range of strategies like expanding, factoring, adding like terms, and applying the distributive property. Understanding how to work with algebraic expressions is fundamental to mastering algebra and advancing in mathematics.

Distributive Property

The distributive property is a cornerstone of algebra, and it's a rule that allows us to multiply one term by a set of terms within parentheses. Essentially, this property tells us how to deal with multiplication when an addition or subtraction is inside parentheses. The formula for the distributive property is \(a(b + c) = ab + ac\).

For instance, if we have a situation like \((z + w)(z + w)(z + w)(z - w)(z - w)\), we can apply the distributive property step-by-step to expand each set of parentheses. By doing that systematically, we keep multiplying and 'distributing' terms across the expression until there are no more parentheses left. This expansion is critical for when we need to simplify expressions or solve for variables.

Combining Like Terms

To combine like terms in algebra, we look for terms in an expression that have the same variable components raised to the same power. For example, \(2x^2\) and \(3x^2\) are like terms, while \(2x^2\) and \(3x^3\) are not, because they have different powers of x.

After expanding expressions using the distributive property, we often end up with an expression that contains many terms. Some of them may appear to be different at first glance, but they could actually be like terms that we can combine. By adding or subtracting the coefficients (the numbers in front of the variables) of these like terms, we simplify the expression, making it easier to understand or solve. For example, \(z^5 + z^5 - z^4w + 3z^4w\) can be simplified to \(2z^5 + 2z^4w\), by combining the like terms.