Problem 83 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(-2 m n^{6}\right)\left(-13 m^{8} n\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(26m^9n^7\).

Step 1: Apply the Negative Rule

When multiplying two negative numbers, the result is positive. Therefore, the expression \((-2mn^6)(-13m^8n)\) becomes \(2mn^6 \cdot 13m^8n\).

Step 2: Multiply the Coefficients

Multiply the numerical coefficients together: \(2 \times 13 = 26\). Thus, the expression becomes \(26m \cdot n^6 \cdot m^8 \cdot n\).

Step 3: Apply the Product of Powers Property to 'm'

Combine the terms involving \(m\) using the product of powers property, which states that \(a^m \cdot a^n = a^{m+n}\). This gives \(m^{1+8} = m^9\).

Step 4: Apply the Product of Powers Property to 'n'

Combine the terms involving \(n\) using the same property. This gives \(n^{6+1} = n^7\).

Step 5: Write the Final Simplified Expression

Substitute \(m^9\) and \(n^7\) back into the expression: \(26m^9n^7\). Thus the simplified expression is \(26m^9n^7\).

Key Concepts

Negative Rule

When working with algebraic expressions, knowing how to handle negative numbers is very important. The negative rule comes in handy particularly in multiplication.
When you multiply two negative numbers together, like in the expression \((-2mn^6)(-13m^8n)\), the negatives cancel each other out. This results in a positive product.

Therefore, multiplying \(-2\) by \(-13\) gives us a positive \(26\). Hence, the expression transforms into \(2mn^6 \cdot 13m^8n\), with no negative signs left. Understanding this rule will make simplifying expressions much smoother.

• Always remember: Negative × Negative = Positive.
• This rule applies to any pair of negative numbers.

Product of Powers Property

The product of powers property is a fundamental concept in algebra that allows you to simplify expressions involving the same base. In the expression \(26m \cdot n^6 \cdot m^8 \cdot n\), we have terms with common bases, such as \(m\) and \(n\).

The property states that when you multiply powers with the same base, you simply add their exponents. Using this property:

• For \(m\), we have \(m^1 \cdot m^8 = m^{1+8} = m^9\).
• For \(n\), similarly, \(n^6 \cdot n^1 = n^{6+1} = n^7\).

Applying this property simplifies the expression significantly. This approach is not only applicable to algebra but also in higher math studies, proving it to be quite useful.

Multivariable Expressions

Multivariable expressions are mathematical expressions that include more than one variable. For example, in the exercise \(2mn^6 \cdot 13m^8n\), the variables are \(m\) and \(n\). Working with multivariable expressions involves combining and simplifying terms that have similar variables or factors.

In order to tackle multivariable expressions effectively:

• Identify and group similar terms to simplify them using algebraic rules, like the product of powers property.
• Keep track of the variables and their exponents carefully, ensuring you apply the correct rules.
• Don't forget to apply the negative rule for coefficients, if necessary.

Simplifying multivariable expressions enhances your understanding of algebra and builds a strong foundation for solving more complex equations in future studies.