Simplify (5x^3)(-2x^4)(-4x)

Solution:

Step 1: Combine the exponents of $x^3$ and $x^4$.

• Begin by grouping $x^3$ and $x^4$. $(5x^3)\cdot (-2x^4)\cdot (-4x)$
• Apply the exponent rule $a^m\cdot a^n=a^{m+n}$. $5x^{3+4}\cdot (-2)\cdot (-4x)$
• Sum the exponents 3 and 4. $5x^7\cdot (-2)\cdot (-4x)$

Step 2: Multiply $x^7$ by $x$.

• Isolate $x$ to combine with $x^7$. $5(x\cdot x^7)\cdot (-2)\cdot (-4)$
• Consider $x$ as $x^1$ and apply the power rule. $5(x^1\cdot x^7)\cdot (-2)\cdot (-4)$
• Combine the exponents 1 and 7. $5x^{1+7}\cdot (-2)\cdot (-4)$
• Sum the exponents 1 and 7. $5x^8\cdot (-2)\cdot (-4)$

Step 3: Multiply the constants $-2$ and $5$.

• Perform the multiplication of constants. $(-2\cdot 5)x^8\cdot (-4)$

Step 4: Multiply the constant $-4$ with the result from Step 3.

• Complete the multiplication with $-4$. $(-4)\cdot (-10)x^8$
• Simplify to get the final result. $40x^8$

Knowledge Notes:

The problem-solving process involves simplifying a product of algebraic expressions using exponent rules. Here are the relevant knowledge points:

1. Multiplication of Like Bases: When multiplying terms with the same base, you add the exponents. This is known as the power rule, which states that $a^m\cdot a^n=a^{m+n}$.

2. Negative Numbers: Multiplying two negative numbers results in a positive number. This is why $-2\cdot -4$ becomes $+8$.

3. Combining Constants: Constants can be multiplied together as a separate operation from variables. In this case, $-2\cdot 5\cdot -4$.

4. Exponent of One: Any variable raised to the power of one is equal to itself, i.e., $x^1=x$.

5. Simplification: The final step in the problem is to simplify the expression by performing all the multiplications, which results in the simplified form $40x^8$.

Understanding these concepts is crucial for simplifying expressions and solving algebraic problems efficiently.