Evaluate The Expression $2^{-2} \times 5^3$, And Give Your Answer As An Integer Or A Reduced Fraction. □ \square □

Mar 12, 2025 by ADMIN 117 views

$Evaluate the Expression: A Step-by-Step Guide to Simplifying $2^{-2} \times 5^3$$

Introduction

In mathematics, expressions involving exponents and multiplication can be complex and challenging to simplify. The given expression $2^{-2} \times 5^3$ requires a thorough understanding of exponent rules and properties to evaluate correctly. In this article, we will break down the expression into manageable parts, apply the necessary rules, and provide a step-by-step guide to simplifying the expression.

Understanding Exponents

Before diving into the expression, it's essential to understand the concept of exponents. An exponent is a small number that is raised to a power, indicating how many times the base number is multiplied by itself. For example, $2^3$ means 2 multiplied by itself 3 times, which equals 8. In the given expression, we have two exponents: $2^{-2}$ and $5^3$.

Simplifying $2^{-2}$

To simplify $2^{-2}$, we need to apply the rule for negative exponents. A negative exponent indicates that the base number is being divided by itself as many times as the exponent. In this case, $2^{-2}$ means 1 divided by 2 squared, which equals 1/4.

Simplifying $5^3$

Now, let's simplify $5^3$. According to the exponent rule, $5^3$ means 5 multiplied by itself 3 times, which equals 125.

Multiplying the Simplified Expressions

Now that we have simplified both expressions, we can multiply them together. $2^{-2} \times 5^3$ equals 1/4 multiplied by 125, which equals 31.25.

Converting the Result to a Reduced Fraction

Since the problem asks for the answer as an integer or a reduced fraction, we need to convert 31.25 to a reduced fraction. To do this, we can express 31.25 as a fraction with a denominator of 4, which equals 125/4.

Conclusion

In conclusion, the expression $2^{-2} \times 5^3$ can be simplified by applying the rules for negative exponents and multiplying the simplified expressions together. The result is 31.25, which can be expressed as a reduced fraction: 125/4.

Frequently Asked Questions

What is the value of $2^{-2}$?
The value of $2^{-2}$ is 1/4.
What is the value of $5^3$?
The value of $5^3$ is 125.
How do you multiply fractions?
To multiply fractions, you multiply the numerators together and the denominators together.
What is the result of multiplying 1/4 and 125?
The result of multiplying 1/4 and 125 is 31.25.

Final Thoughts

Evaluating expressions involving exponents and multiplication requires a thorough understanding of exponent rules and properties. By breaking down the expression into manageable parts and applying the necessary rules, we can simplify the expression and provide a correct answer. In this article, we have provided a step-by-step guide to simplifying the expression $2^{-2} \times 5^3$ and have discussed the importance of understanding exponents and exponent rules.

Additional Resources

For further practice and review, we recommend the following resources:

Khan Academy: Exponents and Exponential Functions
Mathway: Exponent Rules and Properties
Wolfram Alpha: Exponent Rules and Properties

References

"Algebra and Trigonometry" by Michael Sullivan
"Mathematics for the Nonmathematician" by Morris Kline
"The Art of Mathematics" by Tom M. Apostol

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.

Introduction

In our previous article, we explored the concept of evaluating expressions with exponents and multiplication. We simplified the expression $2^{-2} \times 5^3$ and provided a step-by-step guide to understanding exponents and exponent rules. In this article, we will address some of the most frequently asked questions related to evaluating expressions with exponents and multiplication.

Q&A

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

A: The value of $2^{-3}$ is 1/8.

Q: How do you simplify $3^4 \times 2^2$?

A: To simplify $3^4 \times 2^2$, we need to apply the exponent rule for multiplication. $3^4 \times 2^2$ equals 81 multiplied by 4, which equals 324.

Q: What is the value of $5^{-2}$?

A: The value of $5^{-2}$ is 1/25.

Q: How do you evaluate $2^3 \times 3^{-2}$?

A: To evaluate $2^3 \times 3^{-2}$, we need to apply the exponent rule for multiplication. $2^3 \times 3^{-2}$ equals 8 multiplied by 1/9, which equals 8/9.

Q: What is the value of $4^2 \times 3^3$?

A: The value of $4^2 \times 3^3$ is 16 multiplied by 27, which equals 432.

Q: How do you simplify $2^{-4} \times 5^2$?

A: To simplify $2^{-4} \times 5^2$, we need to apply the exponent rule for multiplication. $2^{-4} \times 5^2$ equals 1/16 multiplied by 25, which equals 25/16.

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

A: The value of $3^{-3}$ is 1/27.

Q: How do you evaluate $2^2 \times 3^4$?

A: To evaluate $2^2 \times 3^4$, we need to apply the exponent rule for multiplication. $2^2 \times 3^4$ equals 4 multiplied by 81, which equals 324.