Multiply 4n^4*2n^-3
The problem asks you to perform multiplication on two algebraic terms, specifically 4n raised to the power of 4 (4n^4) and 2n raised to the power of -3 (2n^-3). The operation involves combining these terms by multiplying their numerical coefficients (the numbers 4 and 2) and applying the laws of exponents to the variable 'n' with its given exponents (4 and -3). The laws of exponents will be used to add the powers when the same base is multiplied.
$4 n^{4} \cdot 2 n^{- 3}$
Answer
Solution:
Step 1:
Utilize the commutative property to rearrange the multiplication. $4 \times 2 n^{4} n^{-3}$
Step 2:
Combine like bases by summing their exponents.
Step 2.1:
Rearrange the terms with exponents. $4 \times 2 (n^{-3} n^{4})$
Step 2.2:
Apply the exponent rule $a^{m} \cdot a^{n} = a^{m+n}$ to merge the exponents. $4 \times 2 n^{-3+4}$
Step 2.3:
Calculate the sum of the exponents $-3$ and $4$. $4 \times 2 n^{1}$
Step 3:
Condense the expression $4 \times 2 n^{1}$. $4 \times 2 n$
Step 4:
Perform the multiplication of the constants $4$ and $2$. $8n$
Knowledge Notes:
The problem involves multiplying two algebraic expressions with exponents. The process includes several key knowledge points:
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product. For example, $a \times b = b \times a$.
Combining Like Terms with Exponents: When multiplying terms with the same base, you can add the exponents. This is known as the power rule for exponents, which is expressed as $a^{m} \cdot a^{n} = a^{m+n}$.
Simplifying Expressions: After combining like terms, the expression should be simplified by performing any possible multiplications or other operations.
Negative Exponents: A negative exponent indicates that the base is on the opposite side of a fraction. For example, $a^{-n} = \frac{1}{a^n}$. However, when combining with positive exponents of the same base, you can simply add the exponents as long as the result is not negative.
Final Multiplication: Once the variable part of the expression is simplified, the final step is to multiply any remaining constants to obtain the final simplified product.
In the given problem, these principles are applied to multiply $4n^4$ by $2n^{-3}$, resulting in the simplified expression $8n$.
