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.