Evaluate. Write Your Answer As A Fraction Or Whole Number Without An Exponent.$4^{-2} = \square$

Understanding Negative Exponents

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. A negative exponent, on the other hand, is a shorthand way of expressing a fraction. When we see a negative exponent, such as $4^{-2}$, it can be interpreted as "1 divided by 4 squared" or "1/4^2".

Rewriting Negative Exponents as Fractions

To evaluate $4^{-2}$, we can rewrite it as a fraction using the rule that $a^{-n} = \frac{1}{a^n}$. Applying this rule, we get:

$$4^{-2} = \frac{1}{4^2}$$

Simplifying the Expression

Now that we have rewritten the expression as a fraction, we can simplify it further. To do this, we need to calculate the value of $4^2$. Since $4^2$ means multiplying 4 by itself 2 times, we get:

$$4^2 = 4 \times 4 = 16$$

Substituting the Value

Now that we have the value of $4^2$, we can substitute it back into the original expression:

$$4^{-2} = \frac{1}{16}$$

Conclusion

In conclusion, evaluating a negative exponent is a straightforward process that involves rewriting it as a fraction and simplifying the expression. By applying the rule that $a^{-n} = \frac{1}{a^n}$, we can easily convert negative exponents into fractions. In this case, we found that $4^{-2} = \frac{1}{16}$.

Key Takeaways

• Negative exponents can be rewritten as fractions using the rule $a^{-n} = \frac{1}{a^n}$.
• To simplify an expression with a negative exponent, we need to calculate the value of the base raised to the power of the exponent.
• By following these steps, we can easily evaluate negative exponents and simplify expressions.

Real-World Applications

Negative exponents have numerous real-world applications in fields such as science, engineering, and finance. For example, in physics, negative exponents are used to describe the decay of radioactive materials. In finance, negative exponents are used to calculate compound interest rates.

Common Mistakes to Avoid

When working with negative exponents, it's essential to avoid common mistakes such as:

• Forgetting to rewrite the negative exponent as a fraction.
• Failing to simplify the expression properly.
• Misinterpreting the meaning of negative exponents.

Practice Problems

To reinforce your understanding of negative exponents, try solving the following practice problems:

• Evaluate $3^{-4}$.
• Simplify $2^{-3}$.
• Rewrite $5^{-2}$ as a fraction.

Q&A: Evaluating Negative Exponents

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of expressing a fraction. It can be interpreted as "1 divided by the base raised to the power of the exponent".

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $4^{-2}$ can be rewritten as $\frac{1}{4^2}$.

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

A: To simplify an expression with a negative exponent, follow these steps:

1. Rewrite the negative exponent as a fraction using the rule $a^{-n} = \frac{1}{a^n}$.
2. Calculate the value of the base raised to the power of the exponent.
3. Substitute the value back into the original expression.

Q: What is the value of $3^{-4}$?

A: To evaluate $3^{-4}$, we can rewrite it as a fraction using the rule that $a^{-n} = \frac{1}{a^n}$. This gives us:

$$3^{-4} = \frac{1}{3^4}$$

Next, we need to calculate the value of $3^4$. Since $3^4$ means multiplying 3 by itself 4 times, we get:

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Finally, we can substitute the value back into the original expression:

$$3^{-4} = \frac{1}{81}$$

Q: What is the value of $2^{-3}$?

A: To evaluate $2^{-3}$, we can rewrite it as a fraction using the rule that $a^{-n} = \frac{1}{a^n}$. This gives us:

$$2^{-3} = \frac{1}{2^3}$$

Next, we need to calculate the value of $2^3$. Since $2^3$ means multiplying 2 by itself 3 times, we get:

$$2^3 = 2 \times 2 \times 2 = 8$$

Finally, we can substitute the value back into the original expression:

$$2^{-3} = \frac{1}{8}$$

Q: How do I rewrite $5^{-2}$ as a fraction?

A: To rewrite $5^{-2}$ as a fraction, use the rule that $a^{-n} = \frac{1}{a^n}$. This gives us:

$$5^{-2} = \frac{1}{5^2}$$

Next, we need to calculate the value of $5^2$. Since $5^2$ means multiplying 5 by itself 2 times, we get:

$$5^2 = 5 \times 5 = 25$$

Finally, we can substitute the value back into the original expression:

$$5^{-2} = \frac{1}{25}$$

Q: What are some common mistakes to avoid when working with negative exponents?

A: Some common mistakes to avoid when working with negative exponents include:

• Forgetting to rewrite the negative exponent as a fraction.
• Failing to simplify the expression properly.
• Misinterpreting the meaning of negative exponents.

Q: How can I practice evaluating negative exponents?

A: To practice evaluating negative exponents, try solving the following practice problems:

• Evaluate $3^{-4}$.
• Simplify $2^{-3}$.
• Rewrite $5^{-2}$ as a fraction.

By following these steps and practicing with sample problems, you'll become more confident in your ability to evaluate negative exponents and simplify expressions.