Write The Expression With A Single Exponent.$\left((-2)^7\right)^8=$

Introduction

Exponents are a fundamental concept in mathematics, and understanding how to simplify exponential expressions is crucial for solving various mathematical problems. In this article, we will explore how to simplify the expression $\left((-2)^7\right)^8$ using a single exponent.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^b$ represents $a$ multiplied by itself $b$ times. In the expression $\left((-2)^7\right)^8$, we have two exponents: $7$ and $8$. The inner exponent $7$ represents $-2$ multiplied by itself $7$ times, and the outer exponent $8$ represents the result of the inner exponent multiplied by itself $8$ times.

Simplifying the Expression

To simplify the expression $\left((-2)^7\right)^8$, we need to apply the rule of exponents, which states that $(a^b)^c = a^{bc}$. In this case, we have $(a^7)^8 = a^{7 \cdot 8} = a^{56}$.

Applying the Rule of Exponents

Now, let's apply the rule of exponents to the expression $\left((-2)^7\right)^8$. We have:

$$\left((-2)^7\right)^8 = (-2)^{7 \cdot 8} = (-2)^{56}$$

Evaluating the Expression

To evaluate the expression $(-2)^{56}$, we need to multiply $-2$ by itself $56$ times. However, this is not a feasible task, as it would require a lot of time and effort. Instead, we can use the fact that $(-2)^{56} = ((-2)^2)^{28} = 4^{28}$.

Simplifying the Expression Further

Now, let's simplify the expression $4^{28}$ further. We can rewrite $4$ as $2^2$, so we have:

$$4^{28} = (2^2)^{28} = 2^{2 \cdot 28} = 2^{56}$$

Conclusion

In conclusion, we have simplified the expression $\left((-2)^7\right)^8$ using a single exponent. We applied the rule of exponents, which states that $(a^b)^c = a^{bc}$, to simplify the expression. We also evaluated the expression $(-2)^{56}$ and simplified it further to $2^{56}$. This demonstrates the importance of understanding exponents and how to simplify exponential expressions.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. These include:

• Not applying the rule of exponents: Failing to apply the rule of exponents can lead to incorrect simplifications.
• Not evaluating the expression correctly: Failing to evaluate the expression correctly can lead to incorrect results.
• Not simplifying the expression further: Failing to simplify the expression further can lead to unnecessary complexity.

Tips and Tricks

When simplifying exponential expressions, here are some tips and tricks to keep in mind:

• Use the rule of exponents: The rule of exponents is a powerful tool for simplifying exponential expressions.
• Evaluate the expression correctly: Make sure to evaluate the expression correctly to avoid incorrect results.
• Simplify the expression further: Simplify the expression further to avoid unnecessary complexity.

Real-World Applications

Simplifying exponential expressions has numerous real-world applications. These include:

• Science and engineering: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
• Finance: Exponential expressions are used to calculate interest rates, investment returns, and other financial metrics.
• Computer science: Exponential expressions are used to model algorithmic complexity, data compression, and other computer science concepts.

Conclusion

Q: What is the rule of exponents?

A: The rule of exponents states that $(a^b)^c = a^{bc}$. This means that when we have an exponent raised to another exponent, we can simplify it by multiplying the two exponents together.

Q: How do I apply the rule of exponents?

A: To apply the rule of exponents, simply multiply the two exponents together. For example, if we have $(2^3)^4$, we can simplify it by multiplying the two exponents together: $2^{3 \cdot 4} = 2^{12}$.

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

A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number. For example, $2^3$ represents $2$ multiplied by itself $3$ times, while $2^{-3}$ represents the reciprocal of $2$ multiplied by itself $3$ times.

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

A: To simplify an expression with a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, if we have $2^{-3}$, we can rewrite it as $\frac{1}{2^3}$.

Q: What is the order of operations for simplifying exponential expressions?

A: The order of operations for simplifying exponential expressions is:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Multiply any numbers together.
4. Add or subtract any numbers together.

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

A: To simplify an expression with multiple exponents, we can apply the rule of exponents by multiplying the exponents together. For example, if we have $(2^3)^4 \cdot 2^2$, we can simplify it by multiplying the exponents together: $2^{3 \cdot 4} \cdot 2^2 = 2^{12} \cdot 2^2 = 2^{14}$.

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

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Not applying the rule of exponents.
• Not evaluating the expression correctly.
• Not simplifying the expression further.
• Not following the order of operations.

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

A: To check your work when simplifying exponential expressions, you can:

• Plug in values for the variables to see if the expression simplifies correctly.
• Use a calculator to evaluate the expression.
• Check your work by simplifying the expression in a different way.

Q: What are some real-world applications of simplifying exponential expressions?

A: Some real-world applications of simplifying exponential expressions include:

• Modeling population growth and decay.
• Calculating interest rates and investment returns.
• Modeling chemical reactions and other scientific phenomena.
• Calculating algorithmic complexity and data compression.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for solving various mathematical problems. By applying the rule of exponents, evaluating the expression correctly, and simplifying the expression further, we can simplify complex exponential expressions. This demonstrates the importance of understanding exponents and how to simplify exponential expressions.