Evaluate The Expression:c) ( − 4 C 5 ) 2 \left(-4 C^5\right)^2 ( − 4 C 5 ) 2

Understanding the Problem

When evaluating the expression $\left(-4 c^5\right)^2$, we need to apply the rules of exponents and simplify the expression. The expression involves a negative coefficient and a variable raised to a power, which we will need to handle carefully.

Applying the Power Rule

To evaluate the expression, we will use the power rule of exponents, which states that for any numbers $a$ and $b$ and any integer $n$, $\left(ab\right)^n = a^nb^n$. We will also use the fact that $\left(-a\right)^n = a^n$ when $n$ is even.

Simplifying the Expression

Using the power rule, we can rewrite the expression as $\left(-4 c^5\right)^2 = \left(-4\right)^2 \left(c^5\right)^2$. Now, we can simplify each factor separately.

Simplifying the Coefficient

The coefficient $-4$ is raised to the power of $2$, so we can simplify it as $\left(-4\right)^2 = 16$.

Simplifying the Variable

The variable $c^5$ is also raised to the power of $2$, so we can simplify it as $\left(c^5\right)^2 = c^{10}$.

Combining the Simplified Factors

Now, we can combine the simplified factors to get the final result: $16c^{10}$.

Conclusion

In conclusion, the expression $\left(-4 c^5\right)^2$ simplifies to $16c^{10}$. This result can be verified by plugging in a value for $c$ and evaluating the expression.

Example

Let's plug in $c = 2$ to verify the result. Substituting $c = 2$ into the expression, we get $\left(-4 \cdot 2^5\right)^2 = \left(-4 \cdot 32\right)^2 = \left(-128\right)^2 = 16384$. Now, let's evaluate the simplified expression $16c^{10}$ with $c = 2$: $16 \cdot 2^{10} = 16 \cdot 1024 = 16384$. As we can see, the result is the same, which verifies our simplification.

Key Takeaways

• When evaluating expressions with negative coefficients and variables raised to powers, we need to apply the rules of exponents carefully.
• The power rule of exponents states that for any numbers $a$ and $b$ and any integer $n$, $\left(ab\right)^n = a^nb^n$.
• When a negative coefficient is raised to an even power, it simplifies to a positive coefficient.
• When a variable is raised to a power and then raised to another power, we can simplify it by multiplying the exponents.

Common Mistakes to Avoid

• Failing to apply the power rule of exponents correctly.
• Not simplifying the coefficient and variable separately.
• Not verifying the result by plugging in a value for the variable.

Real-World Applications

• Evaluating expressions with negative coefficients and variables raised to powers is a common task in algebra and calculus.
• Understanding the rules of exponents is essential for solving problems in physics, engineering, and other fields that involve mathematical modeling.

Final Thoughts

Evaluating the expression $\left(-4 c^5\right)^2$ requires careful application of the rules of exponents and simplification of the coefficient and variable. By following the steps outlined in this article, we can simplify the expression and arrive at the final result.

Frequently Asked Questions

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any numbers $a$ and $b$ and any integer $n$, $\left(ab\right)^n = a^nb^n$. This rule allows us to simplify expressions with multiple factors raised to a power.

Q: How do I simplify an expression with a negative coefficient raised to an even power?

A: When a negative coefficient is raised to an even power, it simplifies to a positive coefficient. For example, $\left(-4\right)^2 = 16$.

Q: How do I simplify an expression with a variable raised to a power and then raised to another power?

A: When a variable is raised to a power and then raised to another power, we can simplify it by multiplying the exponents. For example, $\left(c^5\right)^2 = c^{10}$.

Q: What is the difference between $\left(-4 c^5\right)^2$ and $16c^{10}$?

A: $\left(-4 c^5\right)^2$ is the original expression, while $16c^{10}$ is the simplified result. The two expressions are equivalent, but the simplified result is often easier to work with.

Q: How do I verify the result of a simplification?

A: To verify the result of a simplification, plug in a value for the variable and evaluate the expression. For example, if we plug in $c = 2$ into the expression $\left(-4 c^5\right)^2$, we get $\left(-4 \cdot 2^5\right)^2 = \left(-4 \cdot 32\right)^2 = \left(-128\right)^2 = 16384$. Now, let's evaluate the simplified expression $16c^{10}$ with $c = 2$: $16 \cdot 2^{10} = 16 \cdot 1024 = 16384$. As we can see, the result is the same, which verifies our simplification.

Q: What are some common mistakes to avoid when evaluating expressions with negative coefficients and variables raised to powers?

A: Some common mistakes to avoid include:

• Failing to apply the power rule of exponents correctly.
• Not simplifying the coefficient and variable separately.
• Not verifying the result by plugging in a value for the variable.

Q: How do I apply the power rule of exponents in a real-world context?

A: The power rule of exponents is used in a variety of real-world contexts, including:

• Algebra and calculus: Evaluating expressions with negative coefficients and variables raised to powers is a common task in algebra and calculus.
• Physics and engineering: Understanding the rules of exponents is essential for solving problems in physics and engineering that involve mathematical modeling.

Q: What are some tips for simplifying expressions with negative coefficients and variables raised to powers?

A: Some tips for simplifying expressions with negative coefficients and variables raised to powers include:

• Apply the power rule of exponents carefully.
• Simplify the coefficient and variable separately.
• Verify the result by plugging in a value for the variable.

Conclusion

Evaluating expressions with negative coefficients and variables raised to powers requires careful application of the rules of exponents and simplification of the coefficient and variable. By following the steps outlined in this article and avoiding common mistakes, we can simplify expressions and arrive at the final result.