What Is $\left(40 R^6 T^3\right) \div \left(8 R^8 T^3\right$\]? A. 5r^{-2}B. 5r^{-2}tC. 5tD. 5

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: $\left(40 r^6 t^3\right) \div \left(8 r^8 t^3\right)$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Problem

The given problem involves dividing two algebraic expressions: $\left(40 r^6 t^3\right)$ and $\left(8 r^8 t^3\right)$. To simplify this expression, we need to apply the rules of exponents and follow the order of operations.

Step 1: Apply the Quotient Rule for Exponents

When dividing two powers with the same base, we subtract the exponents. In this case, the base is $r$ and $t$. We can rewrite the expression as:

$$\frac{40 r^6 t^3}{8 r^8 t^3} = \frac{40}{8} \cdot \frac{r^6}{r^8} \cdot \frac{t^3}{t^3}$$

Step 2: Simplify the Numerical Coefficient

The numerical coefficient is $\frac{40}{8}$. We can simplify this by dividing both numbers:

$$\frac{40}{8} = 5$$

Step 3: Apply the Quotient Rule for Exponents (continued)

Now, we can apply the quotient rule for exponents to the remaining expression:

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

Step 4: Simplify the Expression

Now that we have simplified the numerical coefficient and the exponents, we can rewrite the expression as:

$$5 \cdot r^{-2} \cdot t^3$$

However, we can further simplify this expression by canceling out the $t^3$ term:

$$5 \cdot r^{-2} \cdot t^3 = 5r^{-2}t^0$$

Since $t^0 = 1$, we can rewrite the expression as:

$$5r^{-2}t^0 = 5r^{-2}$$

Conclusion

In conclusion, the simplified expression is $5r^{-2}$. This is the correct answer to the given problem.

Discussion

Now that we have simplified the expression, let's discuss the different options:

• A. $5r^{-2}$: This is the correct answer.
• B. $5r^{-2}t$: This option is incorrect because we canceled out the $t^3$ term.
• C. $5t$: This option is incorrect because we simplified the expression to $5r^{-2}$.
• D. $5$: This option is incorrect because we simplified the expression to $5r^{-2}$.

Final Answer

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given problem: $\left(40 r^6 t^3\right) \div \left(8 r^8 t^3\right)$. We broke down the solution into manageable steps, making it easy to understand and follow. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

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 moving the base to the other side of the fraction. For example, $a^{-n} = \frac{1}{a^n}$.

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

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

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: To simplify an expression with multiple variables, we can use the quotient rule for exponents to simplify each variable separately. For example, $\frac{a^m b^n}{a^p b^q} = a^{m-p} b^{n-q}$.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same. For example, $x$ is a variable, while $5$ is a constant.

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

A: To simplify an expression with a fraction, we can multiply the numerator and denominator by the same value to eliminate the fraction. For example, $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is $5r^{-2}$.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the order of operations and applying the quotient rule for exponents, we can simplify even the most complex expressions. We hope this Q&A guide has been helpful in answering your questions and providing a better understanding of simplifying algebraic expressions.

Additional Resources

For more information on simplifying algebraic expressions, we recommend the following resources:

• Khan Academy: Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Algebraic Expressions

Final Answer

The final answer is $5r^{-2}$.