Evaluate The Expression: ${ \frac{16^{\frac{5}{4}} \cdot 16 {\frac{1}{4}}}{\left(16 {\frac{1}{2}}\right)^2}= }$

Introduction

In mathematics, the concept of exponents is crucial in solving various types of problems. Exponents are used to represent the power or the index of a number. When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will evaluate the given expression, which involves exponents, and explore the underlying mathematical concepts.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." It represents the product of 2 multiplied by itself 3 times: $2^3 = 2 \cdot 2 \cdot 2 = 8.$

Evaluating the Expression

To evaluate the given expression, we need to apply the rules of exponents. The expression is $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}.$

Simplifying the Expression

We can start by simplifying the numerator. Using the rule of exponents that states $a^m \cdot a^n = a^{m+n},$ we can combine the two terms in the numerator:

$$16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}} = 16^{\frac{5}{4} + \frac{1}{4}} = 16^1 = 16.$$

Simplifying the Denominator

Next, we can simplify the denominator. Using the rule of exponents that states $\left(a^m\right)^n = a^{mn},$ we can rewrite the denominator as:

$$\left(16^{\frac{1}{2}}\right)^2 = 16^{\frac{1}{2} \cdot 2} = 16^1 = 16.$$

Evaluating the Expression

Now that we have simplified the numerator and denominator, we can evaluate the expression:

$$\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2} = \frac{16}{16} = 1.$$

Conclusion

In this article, we evaluated the given expression, which involved exponents, and explored the underlying mathematical concepts. We applied the rules of exponents to simplify the numerator and denominator, and then evaluated the expression to find the final result. The expression $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$ simplifies to 1.

Frequently Asked Questions

• What is the rule of exponents that states $a^m \cdot a^n = a^{m+n}$? The rule of exponents that states $a^m \cdot a^n = a^{m+n}$ is a fundamental property of exponents. It states that when we multiply two numbers with the same base, we can add their exponents.
• What is the rule of exponents that states $\left(a^m\right)^n = a^{mn}$? The rule of exponents that states $\left(a^m\right)^n = a^{mn}$ is another fundamental property of exponents. It states that when we raise a power to a power, we can multiply the exponents.
• How do we simplify the numerator and denominator of the given expression? To simplify the numerator and denominator of the given expression, we can apply the rules of exponents. We can combine the two terms in the numerator using the rule $a^m \cdot a^n = a^{m+n},$ and we can rewrite the denominator using the rule $\left(a^m\right)^n = a^{mn}.$

Final Thoughts

In conclusion, evaluating the expression $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$ requires a deep understanding of the rules of exponents. By applying these rules, we can simplify the numerator and denominator, and then evaluate the expression to find the final result. The expression simplifies to 1, demonstrating the power of exponents in solving mathematical problems.

Additional Resources

• Exponents: A Comprehensive Guide This article provides a comprehensive guide to exponents, including their definition, properties, and rules.
• Evaluating Expressions with Exponents This article provides a step-by-step guide to evaluating expressions with exponents, including examples and practice problems.
• Mathematics: A Subject of Beauty and Wonder This article explores the beauty and wonder of mathematics, including its history, applications, and impact on society.
  

Introduction

In our previous article, we evaluated the expression $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$ and explored the underlying mathematical concepts. In this article, we will answer some frequently asked questions about evaluating expressions with exponents.

Q&A

Q: What is the rule of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: The rule of exponents that states $a^m \cdot a^n = a^{m+n}$ is a fundamental property of exponents. It states that when we multiply two numbers with the same base, we can add their exponents.

Q: What is the rule of exponents that states $\left(a^m\right)^n = a^{mn}$?

A: The rule of exponents that states $\left(a^m\right)^n = a^{mn}$ is another fundamental property of exponents. It states that when we raise a power to a power, we can multiply the exponents.

Q: How do we simplify the numerator and denominator of the given expression?

A: To simplify the numerator and denominator of the given expression, we can apply the rules of exponents. We can combine the two terms in the numerator using the rule $a^m \cdot a^n = a^{m+n},$ and we can rewrite the denominator using the rule $\left(a^m\right)^n = a^{mn}.$

Q: What is the difference between $a^m \cdot a^n$ and $a^{m+n}$?

A: The expressions $a^m \cdot a^n$ and $a^{m+n}$ are equivalent, but they represent different ways of writing the same product. The expression $a^m \cdot a^n$ represents the product of two numbers with the same base, while the expression $a^{m+n}$ represents the result of adding the exponents.

Q: How do we evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, we can use the rule that states $a^{-m} = \frac{1}{a^m}.$ This rule allows us to rewrite negative exponents as fractions.

Q: What is the rule of exponents that states $a^m \div a^n = a^{m-n}$?

A: The rule of exponents that states $a^m \div a^n = a^{m-n}$ is a fundamental property of exponents. It states that when we divide two numbers with the same base, we can subtract their exponents.

Q: How do we simplify expressions with multiple exponents?

A: To simplify expressions with multiple exponents, we can apply the rules of exponents. We can combine the exponents using the rule $a^m \cdot a^n = a^{m+n},$ and we can rewrite the expression using the rule $\left(a^m\right)^n = a^{mn}.$

Conclusion

In this article, we answered some frequently asked questions about evaluating expressions with exponents. We explored the rules of exponents, including the rule $a^m \cdot a^n = a^{m+n}$ and the rule $\left(a^m\right)^n = a^{mn}.$ We also discussed how to simplify expressions with multiple exponents and how to evaluate expressions with negative exponents.

Additional Resources

• Exponents: A Comprehensive Guide This article provides a comprehensive guide to exponents, including their definition, properties, and rules.
• Evaluating Expressions with Exponents This article provides a step-by-step guide to evaluating expressions with exponents, including examples and practice problems.
• Mathematics: A Subject of Beauty and Wonder This article explores the beauty and wonder of mathematics, including its history, applications, and impact on society.

Final Thoughts

In conclusion, evaluating expressions with exponents requires a deep understanding of the rules of exponents. By applying these rules, we can simplify expressions and evaluate them to find the final result. The rules of exponents are a fundamental part of mathematics, and they have many practical applications in science, engineering, and other fields.