Write The Expression As A Single Power Of 7.$\[ \frac{7^8}{7^2} = ? \\]A. \[$7^4\$\] B. \[$7^6\$\] C. \[$7^{10}\$\] D. \[$7^{16}\$\]

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. One of the most common techniques used to simplify exponential expressions is the quotient of powers rule, which states that when we divide two exponential expressions with the same base, we subtract the exponents. In this article, we will explore how to simplify the expression $\frac{7^8}{7^2}$ as a single power of 7.

The Quotient of Powers Rule

The quotient of powers rule is a fundamental concept in algebra that helps us simplify exponential expressions. According to this rule, when we divide two exponential expressions with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base and $m$ and $n$ are the exponents.

Applying the Quotient of Powers Rule

Now that we have understood the quotient of powers rule, let's apply it to the given expression $\frac{7^8}{7^2}$. Using the rule, we can simplify the expression as follows:

$$\frac{7^8}{7^2} = 7^{8-2} = 7^6$$

Therefore, the expression $\frac{7^8}{7^2}$ can be simplified as a single power of 7, which is $7^6$.

Conclusion

In this article, we have learned how to simplify the expression $\frac{7^8}{7^2}$ as a single power of 7 using the quotient of powers rule. This rule is a fundamental concept in algebra that helps us simplify exponential expressions and solve complex problems. By applying this rule, we can simplify the expression and arrive at the final answer, which is $7^6$.

Answer

The final answer is $\boxed{7^6}$.

Why Choose This Answer?

We can choose this answer because it is the result of applying the quotient of powers rule to the given expression. The rule states that when we divide two exponential expressions with the same base, we subtract the exponents. In this case, we have $7^8$ divided by $7^2$, so we subtract the exponents to get $7^6$.

What is the Importance of Simplifying Exponential Expressions?

Simplifying exponential expressions is an important skill in mathematics because it helps us solve complex problems and understand the underlying concepts. By simplifying exponential expressions, we can:

• Solve equations and inequalities
• Graph functions
• Analyze data
• Make predictions

Real-World Applications of Simplifying Exponential Expressions

Simplifying exponential expressions has many real-world applications, including:

• Finance: Exponential expressions are used to calculate interest rates, investments, and loans.
• Science: Exponential expressions are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.

Tips and Tricks for Simplifying Exponential Expressions

Here are some tips and tricks for simplifying exponential expressions:

• Use the quotient of powers rule: When dividing two exponential expressions with the same base, subtract the exponents.
• Use the product of powers rule: When multiplying two exponential expressions with the same base, add the exponents.
• Use the power of a power rule: When raising an exponential expression to a power, multiply the exponents.

Conclusion

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Introduction

In our previous article, we explored how to simplify the expression $\frac{7^8}{7^2}$ as a single power of 7 using the quotient of powers rule. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q&A

Q: What is the quotient of powers rule?

A: The quotient of powers rule is a fundamental concept in algebra that states that when we divide two exponential expressions with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

Q: How do I apply the quotient of powers rule?

A: To apply the quotient of powers rule, simply subtract the exponents of the two exponential expressions with the same base. For example, if we have $\frac{7^8}{7^2}$, we would subtract the exponents to get $7^6$.

Q: What is the product of powers rule?

A: The product of powers rule is another fundamental concept in algebra that states that when we multiply two exponential expressions with the same base, we add the exponents. Mathematically, this can be represented as:

$$a^m \cdot a^n = a^{m+n}$$

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the two exponential expressions with the same base. For example, if we have $7^3 \cdot 7^4$, we would add the exponents to get $7^7$.

Q: What is the power of a power rule?

A: The power of a power rule is a fundamental concept in algebra that states that when we raise an exponential expression to a power, we multiply the exponents. Mathematically, this can be represented as:

$$(a^m)^n = a^{m \cdot n}$$

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, simply multiply the exponents of the two exponential expressions. For example, if we have $(7^3)^4$, we would multiply the exponents to get $7^{12}$.

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

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

• Not using the quotient of powers rule: When dividing two exponential expressions with the same base, make sure to subtract the exponents.
• Not using the product of powers rule: When multiplying two exponential expressions with the same base, make sure to add the exponents.
• Not using the power of a power rule: When raising an exponential expression to a power, make sure to multiply the exponents.

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

A: To check your work when simplifying exponential expressions, simply plug the simplified expression back into the original equation and see if it is true. For example, if we have $\frac{7^8}{7^2} = 7^6$, we can plug $7^6$ back into the original equation to see if it is true.

Conclusion

In conclusion, simplifying exponential expressions is an important skill in mathematics that helps us solve complex problems and understand the underlying concepts. By applying the quotient of powers rule, product of powers rule, and power of a power rule, we can simplify exponential expressions and arrive at the final answer. Remember to avoid common mistakes and check your work to ensure accuracy.

Tips and Tricks for Simplifying Exponential Expressions

Here are some additional tips and tricks for simplifying exponential expressions:

• Use the order of operations: When simplifying exponential expressions, make sure to follow the order of operations (PEMDAS).
• Use a calculator: If you are having trouble simplifying an exponential expression, try using a calculator to check your work.
• Practice, practice, practice: The more you practice simplifying exponential expressions, the more comfortable you will become with the rules and techniques.

Real-World Applications of Simplifying Exponential Expressions

Simplifying exponential expressions has many real-world applications, including:

• Finance: Exponential expressions are used to calculate interest rates, investments, and loans.
• Science: Exponential expressions are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.

Conclusion

In conclusion, simplifying exponential expressions is an important skill in mathematics that helps us solve complex problems and understand the underlying concepts. By applying the quotient of powers rule, product of powers rule, and power of a power rule, we can simplify exponential expressions and arrive at the final answer. Remember to avoid common mistakes and check your work to ensure accuracy.