Simplify (-3a^2b^4)(4a^6n)
The question asks to perform a simplification on the given algebraic expression. The expression consists of two monomials being multiplied together. Each monomial has coefficients (numerical parts), variable 'a' raised to a power, and one of them has variable 'b' raised to a power, while the other one has the variable 'n'. The problem requires applying the rules of exponents and multiplication to combine these monomials into a single, simpler algebraic expression.
$\left(\right. - 3 a^{2} b^{4} \left.\right) \left(\right. 4 a^{6} n \left.\right)$
Answer
Solution:
Step 1: Combine like terms by adding exponents
- Start by focusing on the variable $a$. Multiply $a^2$ by $a^6$.
Step 1.1: Rearrange the expression
- Rearrange to place $a^6$ next to $-3(a^2)$, followed by $b^4(4n)$: $-3(a^6a^2)b^4(4n)$.
Step 1.2: Apply the power rule for exponents
- Use the power rule, which states $a^m \cdot a^n = a^{m+n}$, to combine the exponents of $a$: $-3a^{6+2}b^4(4n)$.
Step 1.3: Perform the addition of exponents
- Add the exponents 6 and 2 for $a$: $-3a^8b^4(4n)$.
Step 2: Multiply the coefficients
- Multiply the coefficients $4$ and $-3$ together: $-12a^8b^4n$.
Knowledge Notes:
The problem requires simplifying a product of two algebraic expressions involving exponents. The process involves the following knowledge points:
Multiplication of like terms: When multiplying terms with the same base, you add the exponents. This is based on the exponentiation rule $a^m \cdot a^n = a^{m+n}$.
Multiplying coefficients: When you multiply terms, you also multiply the coefficients (the numerical parts) together.
Combining terms: Algebraic simplification often requires rearranging terms to combine like terms effectively.
Latex formatting: In the solution, Latex is used to render mathematical expressions, such as $a^{2}$, $a^{6}$, and $-3a^8b^4(4n)$, to make the mathematical notation clear and standardized.
Power rule: This is a specific exponentiation rule used when multiplying terms with the same base. The rule states that to multiply two exponents with the same base, you keep the base and add the exponents.
By understanding and applying these rules, the original expression $(-3a^2b^4)(4a^6n)$ is simplified to $-12a^8b^4n$.
