Which Power Does This Expression Simplify To?${ \left[\left(7 5\right)\left(7 3\right)\right]^{-1} }$A. { \frac{1}{7^2}$}$B. { \frac{1}{7^8}$}$C. ${ 7^4\$} D. ${ 7^n\$}

Understanding Exponential Notation

Exponential notation is a powerful tool used to represent repeated multiplication of a number. In this notation, a number is raised to a power, denoted by an exponent. For example, $7^5$ represents $7$ multiplied by itself $5$ times, or $7 \times 7 \times 7 \times 7 \times 7$. This notation is commonly used in mathematics, particularly in algebra and calculus.

The Power Rule of Exponents

One of the fundamental rules of exponents is the power rule, which states that when a power is raised to another power, the exponents are multiplied. Mathematically, this can be represented as:

$(a^m)^n = a^{mn}$

where $a$ is a non-zero number, $m$ and $n$ are integers, and $a^{mn}$ represents the result of raising $a$ to the power of the product of $m$ and $n$.

Simplifying the Given Expression

The given expression is $\left[\left(7^5\right)\left(7^3\right)\right]^{-1}$. To simplify this expression, we can start by applying the power rule of exponents. Since the expression is raised to the power of $-1$, we can rewrite it as:

$\left[\left(7^5\right)\left(7^3\right)\right]^{-1} = \left(7^{5+3}\right)^{-1}$

Using the power rule, we can simplify the exponent as:

$5+3 = 8$

So, the expression becomes:

$\left(7^8\right)^{-1}$

Applying the Power Rule Again

Now, we can apply the power rule again to simplify the expression further. Since the exponent is raised to the power of $-1$, we can rewrite it as:

$\left(7^8\right)^{-1} = 7^{-8}$

Simplifying the Negative Exponent

A negative exponent can be simplified by taking the reciprocal of the base raised to the positive exponent. In this case, we can rewrite the expression as:

$7^{-8} = \frac{1}{7^8}$

Conclusion

In conclusion, the given expression $\left[\left(7^5\right)\left(7^3\right)\right]^{-1}$ simplifies to $\frac{1}{7^8}$. This result can be obtained by applying the power rule of exponents and simplifying the negative exponent.

Answer

The correct answer is:

B. $\frac{1}{7^8}$

Additional Examples

To further illustrate the concept of simplifying exponential expressions, let's consider a few more examples:

• $\left(2^3\right)^4 = 2^{3 \times 4} = 2^{12}$
• $\left(3^2\right)^{-3} = 3^{-2 \times 3} = 3^{-6} = \frac{1}{3^6}$
• $\left(4^5\right)^{-2} = 4^{-5 \times 2} = 4^{-10} = \frac{1}{4^{10}}$

These examples demonstrate the power rule of exponents and how to simplify negative exponents.

Common Mistakes

When simplifying exponential expressions, it's essential to remember the following common mistakes:

• Not applying the power rule correctly
• Not simplifying negative exponents
• Not using the correct order of operations

By avoiding these mistakes and following the steps outlined in this article, you can simplify exponential expressions with confidence.

Final Thoughts

Q: What is the power rule of exponents?

A: The power rule of exponents states that when a power is raised to another power, the exponents are multiplied. Mathematically, this can be represented as:

$(a^m)^n = a^{mn}$

where $a$ is a non-zero number, $m$ and $n$ are integers, and $a^{mn}$ represents the result of raising $a$ to the power of the product of $m$ and $n$.

Q: How do I simplify a negative exponent?

A: A negative exponent can be simplified by taking the reciprocal of the base raised to the positive exponent. In other words, if you have an expression like $a^{-n}$, you can rewrite it as $\frac{1}{a^n}$.

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

A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents repeated division of a number. For example, $a^3$ represents $a$ multiplied by itself $3$ times, while $a^{-3}$ represents $\frac{1}{a^3}$.

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

A: To apply the power rule, follow these steps:

1. Identify the base and the exponents in the expression.
2. Multiply the exponents together.
3. Simplify the resulting expression.

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

A: When simplifying exponential expressions, follow the order of operations:

1. Evaluate any expressions inside parentheses.
2. Exponents (raise the base to the power of the exponent).
3. Multiplication and division (from left to right).
4. Addition and subtraction (from left to right).

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by applying the power rule. For example, if you have an expression like $(a^m)^n$, you can simplify it by multiplying the exponents together: $(a^m)^n = a^{mn}$.

Q: How do I simplify an expression with a fraction as the base?

A: To simplify an expression with a fraction as the base, follow these steps:

1. Simplify the fraction.
2. Apply the power rule to the simplified fraction.

Q: Can I simplify an expression with a negative base?

A: Yes, you can simplify an expression with a negative base by applying the power rule. For example, if you have an expression like $(-a)^n$, you can simplify it by raising the negative base to the power of $n$.

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 power rule correctly
• Not simplifying negative exponents
• Not using the correct order of operations
• Not simplifying fractions correctly

By avoiding these mistakes and following the steps outlined in this article, you can simplify exponential expressions with confidence.

Conclusion

Simplifying exponential expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the power rule of exponents and how to simplify negative exponents, you can tackle complex problems with ease. Remember to apply the power rule correctly, simplify negative exponents, and use the correct order of operations. With practice and patience, you'll become proficient in simplifying exponential expressions in no time.