Break down the expression into its prime factors

Oct 13, 2025 by windailys.com 21 views

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with exponents, it's essential to understand the rules of exponentiation to simplify complex expressions. In this article, we will focus on simplifying the expression $\left(64 n^{12}\right)^{-\frac{1}{6}}$ using the rules of exponentiation.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$ . When dealing with exponents, we need to follow the rules of exponentiation to simplify expressions.

Simplifying the Expression

To simplify the expression $\left(64 n^{12}\right)^{-\frac{1}{6}}$ , we need to apply the rules of exponentiation. The first step is to break down the expression into its prime factors. We can write $64$ as $2^6$ and $n^{12}$ as $n^{12}$ . Now, we can rewrite the expression as $\left(2^6 n^{12}\right)^{-\frac{1}{6}}$ .

Applying the Power Rule

The power rule states that when we raise a power to another power, we multiply the exponents. In this case, we have $\left(2^6 n^{12}\right)^{-\frac{1}{6}}$ . To simplify this expression, we need to apply the power rule. We can rewrite the expression as $2^{6 \cdot -\frac{1}{6}} n^{12 \cdot -\frac{1}{6}}$ .

Simplifying the Exponents

Now, we can simplify the exponents. We have $2^{6 \cdot -\frac{1}{6}}$ , which equals $2^{-1}$ . Similarly, we have $n^{12 \cdot -\frac{1}{6}}$ , which equals $n^{-2}$ .

Final Simplification

Now, we can simplify the expression further. We have $2^{-1} n^{-2}$ , which can be rewritten as $\frac{1}{2} n^{-2}$ . Since $n^{-2}$ is the same as $\frac{1}{n^2}$ , we can rewrite the expression as $\frac{1}{2} \cdot \frac{1}{n^2}$ .

Conclusion

In conclusion, simplifying the expression $\left(64 n^{12}\right)^{-\frac{1}{6}}$ requires us to apply the rules of exponentiation. We broke down the expression into its prime factors, applied the power rule, and simplified the exponents. Finally, we simplified the expression to $\frac{1}{2} \cdot \frac{1}{n^2}$ .

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with exponents:

Break down the expression into its prime factors: This will help you identify the base and the exponent.
Apply the power rule: This will help you simplify the expression by multiplying the exponents.
Simplify the exponents: This will help you simplify the expression by reducing the exponents.
Use the rules of exponentiation: This will help you simplify the expression by applying the rules of exponentiation.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions with exponents:

Not breaking down the expression into its prime factors: This can lead to incorrect simplification.
Not applying the power rule: This can lead to incorrect simplification.
Not simplifying the exponents: This can lead to incorrect simplification.
Not using the rules of exponentiation: This can lead to incorrect simplification.

Real-World Applications

Simplifying expressions with exponents has many real-world applications. Here are a few examples:

Science and engineering: Simplifying expressions with exponents is crucial in science and engineering, where complex calculations are often required.
Finance: Simplifying expressions with exponents is also important in finance, where complex financial calculations are often required.
Computer programming: Simplifying expressions with exponents is also important in computer programming, where complex algorithms are often required.

Final Thoughts

Introduction

In our previous article, we simplified the expression $\left(64 n^{12}\right)^{-\frac{1}{6}}$ using the rules of exponentiation. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q&A

Q: What is the first step in simplifying an expression with exponents?

A: The first step in simplifying an expression with exponents is to break down the expression into its prime factors. This will help you identify the base and the exponent.

Q: What is the power rule in exponentiation?

A: The power rule in exponentiation states that when we raise a power to another power, we multiply the exponents. For example, $\left(a^m\right)^n = a^{m \cdot n}$ .

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

A: To simplify an expression with a negative exponent, you can rewrite the expression as a fraction. For example, $a^{-n} = \frac{1}{a^n}$ .

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is being multiplied by itself a certain number of times, while a negative exponent indicates that the base is being divided by itself a certain number of times.

Q: Can I simplify an expression with a variable exponent?

A: Yes, you can simplify an expression with a variable exponent by applying the power rule. For example, $\left(a^m\right)^n = a^{m \cdot n}$ .

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can apply the power rule and then simplify the resulting expression. For example, $\left(a^m\right)^n \cdot \left(a^p\right)^q = a^{m \cdot n} \cdot a^{p \cdot q} = a^{m \cdot n + p \cdot q}$ .

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include not breaking down the expression into its prime factors, not applying the power rule, not simplifying the exponents, and not using the rules of exponentiation.

Q: How do I check my work when simplifying expressions with exponents?

A: To check your work when simplifying expressions with exponents, you can plug in some values for the variables and see if the expression simplifies correctly. You can also use a calculator to check your work.