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Answer :
- Multiply the coefficients: $2 \times -3 = -6$.
- Multiply the variable terms using the exponent rule: $x^6 \times x^2 = x^{6+2} = x^8$.
- Combine the results to obtain the simplified expression: $-6x^8$.
- The simplified expression is $\boxed{{-6x^8}}$.
### Explanation
1. Understanding the problem
We are given the expression $2 x^6 \cdot (-3 x^2)$ and we need to simplify it. This involves multiplying the coefficients and combining the variables with their exponents. Remember the rule for multiplying terms with the same base: $x^m \cdot x^n = x^{m+n}$.
2. Multiplying the coefficients
First, let's multiply the coefficients: $2 \cdot (-3) = -6$.
3. Multiplying the variable terms
Next, let's multiply the variable terms: $x^6 \cdot x^2 = x^{6+2} = x^8$.
4. Combining the results
Now, combine the results: $-6x^8$. So, $2 x^6 \cdot (-3 x^2) = -6x^8$.
5. Final Answer
Therefore, the simplified expression is $\boxed{{-6x^8}}$.
### Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the power dissipated in an electrical circuit, you might encounter expressions involving exponents. Simplifying these expressions correctly allows you to accurately determine the power and ensure the circuit operates efficiently and safely. Also, in computer graphics, transformations like scaling involve multiplying coordinates, which can be expressed with exponents, making simplification skills essential for efficient rendering.
- Multiply the variable terms using the exponent rule: $x^6 \times x^2 = x^{6+2} = x^8$.
- Combine the results to obtain the simplified expression: $-6x^8$.
- The simplified expression is $\boxed{{-6x^8}}$.
### Explanation
1. Understanding the problem
We are given the expression $2 x^6 \cdot (-3 x^2)$ and we need to simplify it. This involves multiplying the coefficients and combining the variables with their exponents. Remember the rule for multiplying terms with the same base: $x^m \cdot x^n = x^{m+n}$.
2. Multiplying the coefficients
First, let's multiply the coefficients: $2 \cdot (-3) = -6$.
3. Multiplying the variable terms
Next, let's multiply the variable terms: $x^6 \cdot x^2 = x^{6+2} = x^8$.
4. Combining the results
Now, combine the results: $-6x^8$. So, $2 x^6 \cdot (-3 x^2) = -6x^8$.
5. Final Answer
Therefore, the simplified expression is $\boxed{{-6x^8}}$.
### Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the power dissipated in an electrical circuit, you might encounter expressions involving exponents. Simplifying these expressions correctly allows you to accurately determine the power and ensure the circuit operates efficiently and safely. Also, in computer graphics, transformations like scaling involve multiplying coordinates, which can be expressed with exponents, making simplification skills essential for efficient rendering.
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