Simplify. ( − 4 W 3 X − 6 ) 3 \left(-4 W^3 X^{-6}\right)^3 ( − 4 W 3 X − 6 ) 3

Mar 8, 2025 by ADMIN 79 views

$Simplify.$\left(-4 w^3 x^{-6}\right)^3$$

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this case, we're given the expression $\left(-4 w^3 x^{-6}\right)^3$ and we need to simplify it. To do this, we'll apply the power rule of exponents, which states that for any numbers $a$ and $b$ and any integer $n$ , $\left(ab\right)^n = a^n b^n$ . We'll also use the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$ .

Applying the Power Rule

To simplify the given expression, we'll start by applying the power rule. This means that we'll raise each factor inside the parentheses to the power of $3$ . So, we'll have:

$\left(-4 w^3 x^{-6}\right)^3 = \left(-4\right)^3 \left(w^3\right)^3 \left(x^{-6}\right)^3$

Simplifying the Factors

Now, we'll simplify each factor inside the parentheses. We'll start with the first factor, $-4$ . Raising $-4$ to the power of $3$ gives us:

$\left(-4\right)^3 = -4 \cdot -4 \cdot -4 = -64$

Next, we'll simplify the second factor, $w^3$ . Raising $w^3$ to the power of $3$ gives us:

$\left(w^3\right)^3 = w^3 \cdot w^3 \cdot w^3 = w^9$

Finally, we'll simplify the third factor, $x^{-6}$ . Raising $x^{-6}$ to the power of $3$ gives us:

$\left(x^{-6}\right)^3 = x^{-6} \cdot x^{-6} \cdot x^{-6} = x^{-18}$

Combining the Factors

Now that we've simplified each factor, we can combine them to get the final result. We'll multiply the three factors together:

$\left(-4 w^3 x^{-6}\right)^3 = -64 w^9 x^{-18}$

Understanding the Result

The final result is $-64 w^9 x^{-18}$ . This expression can be rewritten using positive exponents by applying the rule for negative exponents. We'll have:

$-64 w^9 x^{-18} = -64 w^9 \frac{1}{x^{18}}$

Conclusion

In this article, we've simplified the expression $\left(-4 w^3 x^{-6}\right)^3$ using the power rule of exponents. We've applied the rule to each factor inside the parentheses and combined the results to get the final answer. The final result is $-64 w^9 x^{-18}$ , which can be rewritten using positive exponents.

Additional Tips and Tricks

When dealing with exponents, it's essential to remember the rules of exponentiation. Here are some additional tips and tricks to help you simplify expressions:

Power rule: The power rule states that for any numbers $a$ and $b$ and any integer $n$ , $\left(ab\right)^n = a^n b^n$ .
Negative exponents: The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$ .
Exponent rules: Remember that when multiplying numbers with the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .
Simplifying expressions: When simplifying expressions, always look for opportunities to apply the power rule and the rule for negative exponents.

Common Mistakes to Avoid

When dealing with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

Forgetting to apply the power rule: Make sure to apply the power rule to each factor inside the parentheses.
Forgetting to simplify negative exponents: Remember to apply the rule for negative exponents to simplify expressions with negative exponents.
Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.

Real-World Applications

Exponents have many real-world applications. Here are a few examples:

Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and other scientific phenomena.
Finance: Exponents are used to calculate interest rates, investment returns, and other financial calculations.
Computer Science: Exponents are used in algorithms for solving problems in computer science, such as sorting and searching.

Conclusion

In conclusion, simplifying expressions with exponents requires a solid understanding of the rules of exponentiation. By applying the power rule and the rule for negative exponents, we can simplify expressions and arrive at the final result. Remember to always follow the order of operations and avoid common mistakes when dealing with exponents.

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 $n$ , $\left(ab\right)^n = a^n b^n$ . This means that when we raise a product to a power, we can raise each factor to that power.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply raise each factor inside the parentheses to the power of the exponent outside the parentheses. For example, if we have $\left(2x^3\right)^4$ , we would raise each factor to the power of $4$ , resulting in $2^4 x^{3 \cdot 4} = 16x^{12}$ .

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$ . This means that when we have a negative exponent, we can rewrite it as a fraction with the base in the denominator.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we can apply the rule for negative exponents. For example, if we have $x^{-3}$ , we can rewrite it as $\frac{1}{x^3}$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I avoid common mistakes when simplifying expressions with exponents?

A: To avoid common mistakes when simplifying expressions with exponents, make sure to:

Apply the power rule correctly
Simplify negative exponents correctly
Follow the order of operations
Double-check your work to ensure that you have simplified the expression correctly

Q: What are some real-world applications of exponents?

A: Exponents have many real-world applications, including:

Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and other scientific phenomena.
Finance: Exponents are used to calculate interest rates, investment returns, and other financial calculations.
Computer Science: Exponents are used in algorithms for solving problems in computer science, such as sorting and searching.

Q: How can I practice simplifying expressions with exponents?

A: To practice simplifying expressions with exponents, try the following:

Work through practice problems in your textbook or online resources
Use online tools or calculators to check your work and get feedback
Practice simplifying expressions with different types of exponents, such as positive and negative exponents
Try simplifying expressions with multiple exponents, such as $\left(2x^3y^2\right)^4$

Conclusion

In this article, we've answered some frequently asked questions about simplifying expressions with exponents. We've covered topics such as the power rule, negative exponents, the order of operations, and real-world applications of exponents. By following these tips and practicing simplifying expressions with exponents, you'll become more confident and proficient in simplifying expressions with exponents.