Problem 82 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(y^{2} z^{2}\right)\left(y^{15} z^{13}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{17} z^{15}\).

Step 1: Apply the Product of Powers Property

The Product of Powers property states that when multiplying two powers that have the same base, you can add the exponents. The property is given by: \(a^m \times a^n = a^{m+n}\). Apply this to both the \(y\) and \(z\) terms separately.

Step 2: Simplify the Exponents of \(y\)

For \(y\), the expression is \(y^{2} \times y^{15}\). Using the Product of Powers property, add the exponents: \(2 + 15 = 17\). So, \(y^{2} \times y^{15} = y^{17}\).

Step 3: Simplify the Exponents of \(z\)

For \(z\), the expression is \(z^{2} \times z^{13}\). Using the Product of Powers property, add the exponents: \(2 + 13 = 15\). So, \(z^{2} \times z^{13} = z^{15}\).

Step 4: Combine the Simplified Expressions

Combine the simplified expressions from Steps 2 and 3: \(y^{17}\) and \(z^{15}\). The expression simplifies to \(y^{17} z^{15}\).

Key Concepts

Product of Powers

When simplifying mathematical expressions involving exponents, one essential rule to understand is the product of powers property. This rule helps us manage expressions that involve multiplying similar bases with exponents. The product of powers property states that when you multiply two powers with the same base, you simply add their exponents together. Formally, this is written as \(a^m \times a^n = a^{m+n}\).
Let's look at an example to make this clearer. Consider the expression \(x^3 \times x^4\).

• Here, the base is \(x\), and we have exponents 3 and 4.
• Using the product of powers rule, we add the exponents: \(3 + 4 = 7\).
• Therefore, the expression simplifies to \(x^7\).

This property is extremely useful in simplifying expressions like the one in our exercise, where we have terms like \(y^2 z^2 \times y^{15} z^{13}\).
Applying the product of powers, we manage the bases separately, finding the sum of their respective exponents, which makes solving these types of problems much more straightforward.

Simplifying Expressions

Simplifying expressions in algebra often involves making them more straightforward and easier to work with. This process can include combining like terms or applying rules such as the product of powers.
Let's break down how we can simplify expressions using the product of powers property from our problem. In the expression \((y^2 z^2)(y^{15} z^{13})\), we have two components: the \(y\) terms and the \(z\) terms.

• The \(y\) terms \(y^2\) and \(y^{15}\) can be combined by adding their exponents: \(2 + 15 = 17\), resulting in \(y^{17}\).
• The \(z\) terms \(z^2\) and \(z^{13}\) can also be combined: \(2 + 13 = 15\), resulting in \(z^{15}\).

By effectively applying these operations, we simplify the original expression to \(y^{17} z^{15}\), making it simpler and allowing for easier manipulation in other mathematical operations.

Mathematical Operations

Understanding core mathematical operations is vital for grasping more advanced algebra topics. Operations on exponents, such as multiplying powers and simplifying expressions, involve specific rules that allow for clear and logical solutions.
When performing operations with exponents, remember:

• Multiplication of like bases: Add exponents when multiplying powers with the same base. Use the product of powers rule to obtain results quickly and accurately.
• Maintain consistency with bases: Ensure that the bases are identical before applying rules. In our example problem, this means addressing all \(y\)'s together and all \(z\)'s together.
• Organization: Break down the expression into parts, and process each individually as we did with the \(y\) and \(z\) terms in the exercise. Once simplified, combine them for a complete solution.

By following these methods, mathematical operations involving exponents become less daunting, enabling you to simplify expressions efficiently. This preparation sets a sturdy foundation for tackling more complicated algebraic problems.