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Answer :
- Multiply the coefficients: $4 \times -3 \times -7 = 84$.
- Multiply the variables: $x^1 \times x^8 \times x^3 = x^{1+8+3} = x^{12}$.
- Combine the results: $84x^{12}$.
- The product is $\boxed{84 x^{12}}$.
### Explanation
1. Understanding the Problem
Let's break down this problem step by step. We need to multiply three terms together: $(4x)$, $(-3x^8)$, and $(-7x^3)$. Remember that when multiplying terms with exponents, we multiply the coefficients and add the exponents of the variables.
2. Multiplying the Coefficients
First, let's multiply the coefficients: $4 \times -3 \times -7$. The product of $4$ and $-3$ is $-12$. Then, multiplying $-12$ by $-7$ gives us $84$. So, the coefficient of our final term will be $84$.
3. Multiplying the Variables
Now, let's multiply the variable terms: $x \times x^8 \times x^3$. Remember that $x$ is the same as $x^1$. When multiplying variables with exponents, we add the exponents. So, we have $x^{1+8+3} = x^{12}$.
4. Combining the Results
Combining the coefficient and the variable term, we get $84x^{12}$. Therefore, the product of $(4x)(-3x^8)(-7x^3)$ is $84x^{12}$.
### Examples
Understanding how to multiply expressions with exponents is crucial in many areas, such as calculating the area or volume of geometric shapes. For example, if you have a rectangular prism with dimensions $4x$, $-3x^8$, and $-7x^3$, the volume would be the product of these dimensions, which we just calculated to be $84x^{12}$. This type of calculation is also used in physics to determine quantities like kinetic energy or momentum, where variables are often raised to powers.
- Multiply the variables: $x^1 \times x^8 \times x^3 = x^{1+8+3} = x^{12}$.
- Combine the results: $84x^{12}$.
- The product is $\boxed{84 x^{12}}$.
### Explanation
1. Understanding the Problem
Let's break down this problem step by step. We need to multiply three terms together: $(4x)$, $(-3x^8)$, and $(-7x^3)$. Remember that when multiplying terms with exponents, we multiply the coefficients and add the exponents of the variables.
2. Multiplying the Coefficients
First, let's multiply the coefficients: $4 \times -3 \times -7$. The product of $4$ and $-3$ is $-12$. Then, multiplying $-12$ by $-7$ gives us $84$. So, the coefficient of our final term will be $84$.
3. Multiplying the Variables
Now, let's multiply the variable terms: $x \times x^8 \times x^3$. Remember that $x$ is the same as $x^1$. When multiplying variables with exponents, we add the exponents. So, we have $x^{1+8+3} = x^{12}$.
4. Combining the Results
Combining the coefficient and the variable term, we get $84x^{12}$. Therefore, the product of $(4x)(-3x^8)(-7x^3)$ is $84x^{12}$.
### Examples
Understanding how to multiply expressions with exponents is crucial in many areas, such as calculating the area or volume of geometric shapes. For example, if you have a rectangular prism with dimensions $4x$, $-3x^8$, and $-7x^3$, the volume would be the product of these dimensions, which we just calculated to be $84x^{12}$. This type of calculation is also used in physics to determine quantities like kinetic energy or momentum, where variables are often raised to powers.
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