Simplify The Expression: 3 − 1 ÷ 4 − 7 3^{-1} \div 4^{-7} 3 − 1 ÷ 4 − 7

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with exponents and fractions, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore how to simplify the expression $3^{-1} \div 4^{-7}$ using the properties of exponents and fractions.

Understanding Exponents and Fractions

Before we dive into simplifying the expression, let's review the basics of exponents and fractions.

Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ means $2 \times 2 \times 2$, which equals $8$. Exponents can be positive or negative, and they can also be fractional. A negative exponent indicates that we are dealing with a reciprocal, while a fractional exponent indicates that we are dealing with a root.

Fractions

Fractions are a way of representing part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.

Simplifying the Expression

Now that we have a basic understanding of exponents and fractions, let's simplify the expression $3^{-1} \div 4^{-7}$.

Step 1: Rewrite the Expression

To simplify the expression, we can start by rewriting it using the properties of exponents. We can rewrite $3^{-1}$ as $\frac{1}{3}$ and $4^{-7}$ as $\frac{1}{4^7}$.

import math
numerator = 1/3
denominator = 1/(4**7)

print(f"{numerator} ÷ {denominator}")

Step 2: Simplify the Fraction

Now that we have rewritten the expression, we can simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

# Simplify the fraction
simplified_fraction = numerator * (4**7)

print(f"{simplified_fraction}")

Step 3: Evaluate the Expression

Finally, we can evaluate the expression by calculating the value of the simplified fraction.

# Evaluate the expression
result = simplified_fraction

print(f"The final answer is {result}.")

Conclusion

Simplifying the expression $3^{-1} \div 4^{-7}$ requires a deep understanding of exponents and fractions. By rewriting the expression using the properties of exponents and simplifying the fraction, we can arrive at the final answer. In this article, we have demonstrated how to simplify the expression using Python code and mathematical reasoning.

Additional Tips and Tricks

• When dealing with negative exponents, remember that they indicate a reciprocal.
• When dealing with fractional exponents, remember that they indicate a root.
• When simplifying fractions, remember to multiply the numerator by the reciprocal of the denominator.
• When evaluating expressions, remember to calculate the value of the simplified fraction.

Frequently Asked Questions

• Q: What is the difference between a negative exponent and a fractional exponent? A: A negative exponent indicates a reciprocal, while a fractional exponent indicates a root.
• Q: How do I simplify a fraction? A: To simplify a fraction, multiply the numerator by the reciprocal of the denominator.
• Q: What is the final answer to the expression $3^{-1} \div 4^{-7}$? A: The final answer is $\frac{4^7}{3}$.

Final Answer

The final answer is $\boxed{\frac{4^7}{3}}$.

Introduction

In our previous article, we explored how to simplify the expression $3^{-1} \div 4^{-7}$ using the properties of exponents and fractions. In this article, we will answer some frequently asked questions about simplifying expressions with exponents and fractions.

Q&A

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

A: A negative exponent indicates a reciprocal, while a fractional exponent indicates a root. For example, $2^{-3}$ means $\frac{1}{2^3}$, while $2^{1/2}$ means $\sqrt{2}$.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, you can multiply the numerator by the reciprocal of the denominator. For example, $\frac{2^3}{2^2}$ can be simplified by multiplying the numerator by the reciprocal of the denominator: $\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$.

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

A: When multiplying exponents with the same base, you can add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

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

A: To simplify an expression with a negative exponent in the denominator, you can rewrite the expression using the properties of exponents. For example, $\frac{1}{2^{-3}}$ can be rewritten as $2^3$.

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

A: When dividing exponents with the same base, you can subtract the exponents. For example, $2^5 \div 2^3 = 2^{5-3} = 2^2$.

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

A: To simplify an expression with a fractional exponent, you can rewrite the expression using the properties of exponents. For example, $2^{1/2}$ can be rewritten as $\sqrt{2}$.

Q: What is the final answer to the expression $3^{-1} \div 4^{-7}$?

A: The final answer is $\frac{4^7}{3}$.

Additional Tips and Tricks

• When dealing with negative exponents, remember that they indicate a reciprocal.
• When dealing with fractional exponents, remember that they indicate a root.
• When simplifying fractions, remember to multiply the numerator by the reciprocal of the denominator.
• When evaluating expressions, remember to calculate the value of the simplified fraction.

Conclusion

Simplifying expressions with exponents and fractions can be challenging, but with practice and patience, you can master the skills. In this article, we have answered some frequently asked questions about simplifying expressions with exponents and fractions. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{\frac{4^7}{3}}$.