Evaluate. Write Your Answer As A Fraction Or Whole Number Without Exponents. 7 − 3 = 7^{-3} = 7 − 3 = □ \square □

Understanding the Problem

When dealing with negative exponents, it's essential to understand the concept of reciprocals and how they relate to the base number. In this case, we're given the expression $7^{-3}$, and we're asked to evaluate it without using exponents.

The Concept of Negative Exponents

A negative exponent indicates that we're dealing with the reciprocal of the base number raised to a positive power. In other words, $a^{-n} = \frac{1}{a^n}$. This concept is crucial in simplifying expressions with negative exponents.

Applying the Concept to the Given Expression

Using the concept of negative exponents, we can rewrite the expression $7^{-3}$ as $\frac{1}{7^3}$. This is because the negative exponent indicates that we're dealing with the reciprocal of the base number raised to a positive power.

Evaluating the Expression

Now that we've rewritten the expression as $\frac{1}{7^3}$, we can evaluate it by calculating the value of $7^3$. $7^3 = 7 \times 7 \times 7 = 343$. Therefore, $\frac{1}{7^3} = \frac{1}{343}$.

Conclusion

In conclusion, the value of $7^{-3}$ is $\frac{1}{343}$. This is because the negative exponent indicates that we're dealing with the reciprocal of the base number raised to a positive power.

Additional Examples

To further illustrate the concept of negative exponents, let's consider a few additional examples.

Example 1

Evaluate the expression $2^{-4}$.

Using the concept of negative exponents, we can rewrite the expression as $\frac{1}{2^4}$. Evaluating the expression, we get $\frac{1}{2^4} = \frac{1}{16}$.

Example 2

Evaluate the expression $3^{-2}$.

Using the concept of negative exponents, we can rewrite the expression as $\frac{1}{3^2}$. Evaluating the expression, we get $\frac{1}{3^2} = \frac{1}{9}$.

Summary

In this article, we've evaluated the expression $7^{-3}$ and found that its value is $\frac{1}{343}$. We've also explored the concept of negative exponents and how they relate to the base number. By understanding this concept, we can simplify expressions with negative exponents and evaluate them easily.

Key Takeaways

• Negative exponents indicate that we're dealing with the reciprocal of the base number raised to a positive power.
• We can rewrite expressions with negative exponents as fractions using the concept of reciprocals.
• Evaluating expressions with negative exponents involves calculating the value of the base number raised to a positive power.

Final Thoughts

In conclusion, the concept of negative exponents is a powerful tool in simplifying expressions and evaluating them easily. By understanding this concept, we can tackle a wide range of mathematical problems and arrive at accurate solutions.

Understanding Negative Exponents

In our previous article, we explored the concept of negative exponents and how they relate to the base number. We also evaluated the expression $7^{-3}$ and found that its value is $\frac{1}{343}$. In this article, we'll answer some frequently asked questions about negative exponents and provide additional examples to help solidify your understanding.

Q&A

Q1: What is a negative exponent?

A1: A negative exponent indicates that we're dealing with the reciprocal of the base number raised to a positive power. In other words, $a^{-n} = \frac{1}{a^n}$.

Q2: How do I evaluate an expression with a negative exponent?

A2: To evaluate an expression with a negative exponent, you can rewrite it as a fraction using the concept of reciprocals. For example, $7^{-3}$ can be rewritten as $\frac{1}{7^3}$.

Q3: What is the difference between a negative exponent and a positive exponent?

A3: A negative exponent indicates that we're dealing with the reciprocal of the base number raised to a positive power, while a positive exponent indicates that we're dealing with the base number raised to a positive power. For example, $7^{-3}$ is the reciprocal of $7^3$, while $7^3$ is the base number raised to a positive power.

Q4: Can I simplify expressions with negative exponents?

A4: Yes, you can simplify expressions with negative exponents by rewriting them as fractions using the concept of reciprocals. For example, $2^{-4}$ can be rewritten as $\frac{1}{2^4}$.

Q5: How do I handle negative exponents with fractions?

A5: When dealing with negative exponents and fractions, you can rewrite the fraction as a product of two fractions using the concept of reciprocals. For example, $\frac{1}{2^{-3}}$ can be rewritten as $\frac{2^3}{1}$.

Q6: Can I use negative exponents with variables?

A6: Yes, you can use negative exponents with variables. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Q7: How do I evaluate expressions with negative exponents and variables?

A7: To evaluate expressions with negative exponents and variables, you can rewrite the expression as a fraction using the concept of reciprocals. For example, $x^{-3}$ can be rewritten as $\frac{1}{x^3}$.

Additional Examples

To further illustrate the concept of negative exponents, let's consider a few additional examples.

Example 1

Evaluate the expression $4^{-2}$.

Using the concept of negative exponents, we can rewrite the expression as $\frac{1}{4^2}$. Evaluating the expression, we get $\frac{1}{4^2} = \frac{1}{16}$.

Example 2

Evaluate the expression $5^{-1}$.

Using the concept of negative exponents, we can rewrite the expression as $\frac{1}{5^1}$. Evaluating the expression, we get $\frac{1}{5^1} = \frac{1}{5}$.

Example 3

Evaluate the expression $x^{-4}$.

Using the concept of negative exponents, we can rewrite the expression as $\frac{1}{x^4}$. Evaluating the expression, we get $\frac{1}{x^4}$.

Summary

In this article, we've answered some frequently asked questions about negative exponents and provided additional examples to help solidify your understanding. By understanding the concept of negative exponents, you can simplify expressions and evaluate them easily.

Key Takeaways

• Negative exponents indicate that we're dealing with the reciprocal of the base number raised to a positive power.
• We can rewrite expressions with negative exponents as fractions using the concept of reciprocals.
• Evaluating expressions with negative exponents involves calculating the value of the base number raised to a positive power.
• Negative exponents can be used with variables.
• We can simplify expressions with negative exponents by rewriting them as fractions using the concept of reciprocals.

Final Thoughts

In conclusion, the concept of negative exponents is a powerful tool in simplifying expressions and evaluating them easily. By understanding this concept, you can tackle a wide range of mathematical problems and arrive at accurate solutions.