Problem 97 - Algebra A Combined Function

Question

Simplify each exponential expression. $$ \left(-3 x^{2} y^{3} z^{5}\right)\left(20 x^{5} y^{7}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-60 x^7 y^{10} z^5\).

Step 1: Distribute the Coefficients

First, take the coefficients from each term inside the parentheses and multiply them together. The coefficients are \(-3\) and \(20\). Multiply: \(-3 \times 20 = -60\).

Step 2: Simplify the "x" Terms

Next, simplify the terms involving \(x\). In the expression, we have \(x^2\) from the first group and \(x^5\) from the second. Add the exponents since we are multiplying: \(x^{2+5} = x^7\).

Step 3: Simplify the "y" Terms

Simplify the \(y\) terms by adding their exponents. We have \(y^3\) and \(y^7\). Add them up: \(y^{3+7} = y^{10}\).

Step 4: Simplify the "z" Terms

For the \(z\) terms, you only have \(z^5\) in the first group and no \(z\) term in the second, so the \(z\) term remains \(z^5\).

Step 5: Combine All Parts

Combine all the simplified parts to write the fully simplified expression: \(-60 x^7 y^{10} z^5\).

Key Concepts

Distributive Property

The distributive property is one of mathematics' most fundamental and useful properties. It's often used to simplify expressions that involve parentheses. When you apply the distributive property, you multiply each term inside a set of parentheses by the factor outside.

In our given expression, \[(-3x^{2}y^{3}z^{5})(20x^{5}y^{7})\]applying the distributive property involves multiplying the coefficients \(-3\) and \(20\), and then distributing this multiplication across the rest of the expression.

You take each variable term with its exponent from both elements in the parentheses and prepare them for further operations, such as adding the exponents when dealing with the same base. Utilizing the distributive property effectively sets the stage for combining like terms and simplifying the whole expression.

Multiplication of Exponents

Multiplication of exponents is crucial when simplifying expressions with the same base variable. This rule states that when you multiply like bases, you add their exponents.

In the expression \((-3 x^{2} y^{3} z^{5}) (20 x^{5} y^{7})\), we encounter this rule with the terms involving \(x\) and \(y\). For the \(x\) terms, we have \(x^{2}\) and \(x^{5}\). By applying the rule of multiplication of exponents, you add the exponents: \(x^{2+5} = x^{7}\).

Similarly, for the \(y\) terms, you have \(y^{3}\) and \(y^{7}\), adding the exponents gives you \(y^{3+7} = y^{10}\). This method allows you to manage expressions with exponents quickly and efficiently, turning potentially complicated expressions into workable ones.

Combining Like Terms

Combining like terms is an essential step that simplifies algebraic expressions, especially in exponentiation problems. Like terms are terms that contain the same variables raised to the same powers, and by combining them, we reduce an equation to its simplest form.

In our problem, combining like terms involves recognizing all instances of \(x\) and \(y\) from both parts of the expression and then summing their exponents as described previously. Since the terms \(z^{5}\) appear only once in one group of the expression, they do not need to be combined but are included in the final simplified expression.

Therefore, the expression turns into a cleaner, more concise form: \(-60 x^{7} y^{10} z^{5}\), exhibiting the efficiency and clarity that comes from combining like terms.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operational symbols that represent a specific value. They are fundamental to many areas of mathematics and allow for the generalization of arithmetic operations, enabling us to simplify and solve problems systematically.

The original exercise expression \((-3x^{2}y^{3}z^{5})(20x^{5}y^{7})\) is a prime example of an algebraic expression at work. It necessitates using algebraic rules, such as the distributive property and the multiplication of exponents, to be simplified.

These expressions can model real-world situations and embody the potential to transform complex problems into simpler components, allowing for easier manipulation and understanding of mathematical relationships.