Which Power Does This Expression Simplify To? ( ( 7 5 ) ( 7 3 ) ) 4 \left(\left(7^5\right)\left(7^3\right)\right)^4 ( ( 7 5 ) ( 7 3 ) ) 4 A. 1 7 2 \frac{1}{7^2} 7 2 1 ​ B. 1 7 8 \frac{1}{7^8} 7 8 1 ​ C. 7 4 7^4 7 4 D. 7 ′ ′ 7^{\prime \prime} 7 ′′

Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will explore the simplification of exponential expressions, focusing on the given expression $\left(\left(7^5\right)\left(7^3\right)\right)^4$. We will break down the expression, apply the rules of exponents, and arrive at the simplified form.

The Rules of Exponents

Before we dive into the simplification process, let's review the rules of exponents. The rules state that when we multiply two exponential expressions with the same base, we add their exponents. Conversely, when we divide two exponential expressions with the same base, we subtract their exponents. Additionally, when we raise an exponential expression to a power, we multiply the exponents.

Simplifying the Given Expression

Let's start by simplifying the given expression $\left(\left(7^5\right)\left(7^3\right)\right)^4$. We can begin by applying the rule of exponents that states when we multiply two exponential expressions with the same base, we add their exponents.

$\left(\left(7^5\right)\left(7^3\right)\right)^4 = \left(7^{5+3}\right)^4 = \left(7^8\right)^4$

Now, we can apply the rule of exponents that states when we raise an exponential expression to a power, we multiply the exponents.

$\left(7^8\right)^4 = 7^{8 \cdot 4} = 7^{32}$

Therefore, the simplified form of the given expression is $7^{32}$.

Comparing the Simplified Form with the Options

Now that we have simplified the given expression, let's compare it with the options provided.

A. $\frac{1}{7^2}$ B. $\frac{1}{7^8}$ C. $7^4$ D. $7^{32}$

Based on our simplification, we can see that the correct answer is D. $7^{32}$.

Conclusion

In this article, we explored the simplification of exponential expressions, focusing on the given expression $\left(\left(7^5\right)\left(7^3\right)\right)^4$. We applied the rules of exponents, broke down the expression, and arrived at the simplified form. We also compared the simplified form with the options provided and determined that the correct answer is D. $7^{32}$.

Key Takeaways

• Exponential expressions are a fundamental concept in mathematics.
• The rules of exponents state that when we multiply two exponential expressions with the same base, we add their exponents.
• When we raise an exponential expression to a power, we multiply the exponents.
• Simplifying exponential expressions involves applying the rules of exponents and breaking down the expression.

Final Thoughts

Q: What is the rule for multiplying exponential expressions with the same base?

A: When we multiply two exponential expressions with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the rule for dividing exponential expressions with the same base?

A: When we divide two exponential expressions with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the rule for raising an exponential expression to a power?

A: When we raise an exponential expression to a power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression like $\left(\left(7^5\right)\left(7^3\right)\right)^4$?

A: To simplify this expression, we can start by applying the rule for multiplying exponential expressions with the same base. This gives us $\left(7^{5+3}\right)^4 = \left(7^8\right)^4$. Then, we can apply the rule for raising an exponential expression to a power, which gives us $7^{8 \cdot 4} = 7^{32}$.

Q: What is the difference between $a^m$ and $a^{-m}$?

A: $a^m$ represents a positive exponent, while $a^{-m}$ represents a negative exponent. When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I simplify an expression like $\frac{1}{a^m}$?

A: To simplify this expression, we can rewrite it as $a^{-m}$. This is because the reciprocal of $a^m$ is $a^{-m}$.

Q: What is the rule for simplifying expressions with zero exponents?

A: When we have an expression with a zero exponent, we can simplify it by setting the base equal to 1. For example, $a^0 = 1$.

Q: How do I simplify an expression like $(a^m)^0$?

A: To simplify this expression, we can apply the rule for raising an exponential expression to a power. This gives us $a^{m \cdot 0} = a^0 = 1$.

Q: What is the rule for simplifying expressions with negative exponents?

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

Q: How do I simplify an expression like $\frac{1}{a^{-m}}$?

A: To simplify this expression, we can rewrite it as $a^m$. This is because the reciprocal of $a^{-m}$ is $a^m$.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying exponential expressions. We have covered topics such as multiplying and dividing exponential expressions, raising an exponential expression to a power, and simplifying expressions with zero and negative exponents. By understanding these rules and applying them to different types of expressions, we can simplify complex exponential expressions and make informed decisions.