Multiply 5a*2a*(a^3-3a^2+a)
The problem provided is a mathematical expression that requires simplifying the product of three terms involving algebraic variables and exponents. The task is to multiply a binomial expression, which is raised to the power of three and includes both linear and quadratic terms, by two monomials. The first monomial is "5a" and the second one is "2a". The question is asking to perform the multiplication following the rules of algebra and to combine like terms where possible to obtain a simplified expression.
$5 a \cdot 2 a \cdot \left(\right. a^{3} - 3 a^{2} + a \left.\right)$
Answer
Solution:
Step 1: Combine like terms by using the exponent addition rule.
Step 1.1: Position $a$ with $5(a \cdot a) \cdot 2 \cdot (a^3 - 3a^2 + a)$
Step 1.2: Compute $a \cdot a$ as $5a^2 \cdot 2 \cdot (a^3 - 3a^2 + a)$, resulting in $5a^2 \cdot 2 \cdot (a^3 - 3a^2 + a)$
Step 2: Calculate the product of $2$ and $5$ to get $10a^2 \cdot (a^3 - 3a^2 + a)$
Step 3: Distribute $10a^2$ across the polynomial $a^3 - 3a^2 + a$.
Step 4: Begin simplification process.
Step 4.1: Combine $a^2$ and $a^3$ by summing their exponents.
Step 4.1.1: Rearrange to $10(a^3a^2) + 10a^2(-3a^2) + 10a^2a$
Step 4.1.2: Apply the exponent rule $a^ma^n = a^{m+n}$ to merge exponents, resulting in $10a^{3+2} + 10a^2(-3a^2) + 10a^2a$
Step 4.1.3: Add exponents to get $10a^5 + 10a^2(-3a^2) + 10a^2a$
Step 4.2: Use the commutative property to rearrange multiplication as $10a^5 + 10 \cdot -3a^2a^2 + 10a^2a$
Step 4.3: Add exponents for $a^2$ and $a$.
Step 4.3.1: Reposition $a$ to get $10a^5 + 10 \cdot -3a^2a^2 + 10(a \cdot a^2)$
Step 4.3.2: Calculate the product of $a$ and $a^2$.
Step 4.3.2.1: Express $a$ as $a^1$ to get $10a^5 + 10 \cdot -3a^2a^2 + 10(a^1a^2)$
Step 4.3.2.2: Apply the exponent rule $a^ma^n = a^{m+n}$ to combine exponents, yielding $10a^5 + 10 \cdot -3a^2a^2 + 10a^{1+2}$
Step 4.3.3: Sum the exponents to find $10a^5 + 10 \cdot -3a^2a^2 + 10a^3$
Step 5: Simplify each term individually.
Step 5.1: Merge $a^2$ with $a^2$ by adding their exponents.
Step 5.1.1: Rearrange to $10a^5 + 10 \cdot -3(a^2a^2) + 10a^3$
Step 5.1.2: Apply the exponent rule $a^ma^n = a^{m+n}$ to combine exponents, resulting in $10a^5 + 10 \cdot -3a^{2+2} + 10a^3$
Step 5.1.3: Add exponents to get $10a^5 + 10 \cdot -3a^4 + 10a^3$
Step 5.2: Multiply $10$ by $-3$ to finalize the expression as $10a^5 - 30a^4 + 10a^3$
Knowledge Notes:
Exponent Rules: When multiplying like bases, add the exponents ($a^m \cdot a^n = a^{m+n}$). This is known as the Product of Powers rule.
Distributive Property: When multiplying a monomial by a polynomial, multiply the monomial by each term in the polynomial separately ($a(b + c) = ab + ac$).
Commutative Property of Multiplication: The order in which two numbers are multiplied does not change the product ($ab = ba$).
Combining Like Terms: Terms with the same variable parts and exponents can be combined by adding or subtracting the coefficients.
Simplification: The process of rewriting an expression in a simpler or more concise form, often by combining like terms and applying arithmetic operations.
