Multiply 5a*2a*(a^3-3a^2+a)

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 $10(a^3{a}^2)+10a^2(-3a^2)+10a^{2}a$

Step 4.1.2: Apply the exponent rule $a^m{a}^n=a^{m+n}$ to merge exponents, resulting in $10a^{3+2}+10a^2(-3a^2)+10a^{2}a$

Step 4.1.3: Add exponents to get $10a^5+10a^2(-3a^2)+10a^{2}a$

Step 4.2: Use the commutative property to rearrange multiplication as $10a^5+10\cdot -3a^2{a}^2+10a^{2}a$

Step 4.3: Add exponents for $a^2$ and $a$.

Step 4.3.1: Reposition $a$ to get $10a^5+10\cdot -3a^2{a}^2+10(a\cdot a^2)$

Step 4.3.2: Calculate the product of $a$ and $a^2$.

Step 4.3.2.1: Express $a$ as $a^1$ to get $10a^5+10\cdot -3a^2{a}^2+10(a^1{a}^2)$

Step 4.3.2.2: Apply the exponent rule $a^m{a}^n=a^{m+n}$ to combine exponents, yielding $10a^5+10\cdot -3a^2{a}^2+10a^{1+2}$

Step 4.3.3: Sum the exponents to find $10a^5+10\cdot -3a^2{a}^2+10a^3$

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 $10a^5+10\cdot -3(a^2{a}^2)+10a^3$

Step 5.1.2: Apply the exponent rule $a^m{a}^n=a^{m+n}$ 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:

1. 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.

2. Distributive Property: When multiplying a monomial by a polynomial, multiply the monomial by each term in the polynomial separately ($a(b+c)=ab+ac$).

3. Commutative Property of Multiplication: The order in which two numbers are multiplied does not change the product ($ab=ba$).

4. Combining Like Terms: Terms with the same variable parts and exponents can be combined by adding or subtracting the coefficients.

5. Simplification: The process of rewriting an expression in a simpler or more concise form, often by combining like terms and applying arithmetic operations.