Problem 29 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(-8 m n^{6}\right)\left(9 m^{2} n^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-72m^3n^8\).

Step 1: Identify Parts for Product Rule

The expression to simplify is \((-8mn^6)(9m^2n^2)\). It consists of two terms: \(-8mn^6\) and \(9m^2n^2\).

Step 2: Apply the Commutative Property

Use the commutative property of multiplication to rearrange and group like terms together. The expression becomes \((-8 \cdot 9)(m \cdot m^2)(n^6 \cdot n^2)\).

Step 3: Multiply the Coefficients

Multiply the numerical coefficients: \(-8 \times 9 = -72\).

Step 4: Use the Product Rule for Powers of m

Apply the product rule for exponents to the \(m\) terms: \(m \cdot m^2 = m^{1+2} = m^3\).

Step 5: Use the Product Rule for Powers of n

Apply the product rule for exponents to the \(n\) terms: \(n^6 \cdot n^2 = n^{6+2} = n^8\).

Step 6: Combine All Parts Together

Combine the results to simplify the expression: \(-72m^3n^8\).

Key Concepts

Product Rule

The product rule is a fundamental concept in algebra that helps us simplify expressions involving exponents. When you have the same base multiplied together, you can add the exponents. For example, if you have bases of the same type, such as \(x^a\) and \(x^b\), the product rule states: \[ x^a \cdot x^b = x^{a + b} \] This simplifies multiplication and lets you easily manage terms involving powers. In the exercise, the product rule is applied to both \(m\) and \(n\) terms. For \(m\), \(m\cdot m^2\) uses: \[ m^{1} \cdot m^{2} = m^{1+2} = m^3 \] And for \(n\), \(n^6 \cdot n^2\) becomes: \[ n^{6} \cdot n^{2} = n^{6+2} = n^8 \] It simplifies working with powers and streamlines calculation.

Exponents

Exponents represent repeated multiplication of a number by itself. For instance, \(n^6\) means \(n\) multiplied by itself six times. They provide a concise way to express large numbers or complex multiplications. Key points about exponents:

• Any number to the power of one is the number itself, \(x^1 = x\).
• Zero exponent means any number is equal to one, \(x^0 = 1\) (for \(x eq 0\)).
• Multiplying same bases adds the exponents.
• Division works similarly but subtracts the exponents.

In our worked problem, exponents let us condense terms like \(m \cdot m^2\)into \(m^3\), saving time and reducing complexity.

Commutative Property

The commutative property is a basic mathematical concept that states you can change the order of numbers in addition or multiplication without changing the result. For multiplication, it is expressed as: \[ a \cdot b = b \cdot a \] In the solution, it allowed us to rearrange terms for easier manipulation: \((-8 \cdot 9)(m \cdot m^2)(n^6 \cdot n^2)\). This makes it more straightforward to group and apply rules such as the product rule. Utilizing this property can simplify calculations and rearrange expressions for clearer problem-solving. It's particularly useful when dealing with complex algebraic expressions, enabling streamlined grouping of like terms.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form. This makes them easier to understand and work with, especially in algebra.The process:

• Identify like terms, such as similar variables or coefficients.
• Rearrange using properties like the commutative property.
• Apply algebraic rules, such as the product rule for exponents.
• Simplify step by step until reaching the simplest expression.

In our exercise, the expression \((-8mn^6)(9m^2n^2)\) simplifies to \(-72m^3n^8\) by following these steps. Each step breaks down complex mathematics into logical, smaller tasks, focusing on grouping and reducing the problem into an easy, manageable form.