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

Introduction

In mathematics, expressions involving exponents and fractions can be complex and challenging to simplify. The given expression, $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$, is a perfect example of such a problem. In this article, we will evaluate the expression, simplify the exponents, and provide a step-by-step solution to arrive at the final answer.

Understanding Exponents and Fractions

Before we dive into the solution, let's briefly review the concepts of exponents and fractions. Exponents are a shorthand way of representing repeated multiplication. For example, $a^b$ means $a$ multiplied by itself $b$ times. Fractions, on the other hand, are a way of representing a part of a whole. In this case, we have a fraction with exponents in the numerator and denominator.

Simplifying the Exponents

To simplify the expression, we need to start by simplifying the exponents. We can use the property of exponents that states $a^b \cdot a^c = a^{b+c}$. Using this property, we can rewrite the numerator as:

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

Simplifying the Denominator

Next, we need to simplify the denominator. We can use the property of exponents that states $(a^b)^c = a^{bc}$. Using this property, 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 conclusion, we have successfully evaluated the expression and simplified the exponents and fractions. The final answer is 1. This problem demonstrates the importance of understanding and applying the properties of exponents and fractions in order to simplify complex expressions.

Additional Tips and Tricks

• When simplifying exponents, always look for opportunities to combine like terms.
• When simplifying fractions, always look for opportunities to cancel out common factors.
• Practice, practice, practice! The more you practice simplifying exponents and fractions, the more comfortable you will become with the concepts.

Real-World Applications

• Simplifying exponents and fractions is an essential skill in many real-world applications, including science, engineering, and finance.
• Understanding and applying the properties of exponents and fractions can help you to solve complex problems and make informed decisions.
• In science and engineering, simplifying exponents and fractions can help you to model and analyze complex systems.

Common Mistakes to Avoid

• When simplifying exponents, always be careful to combine like terms correctly.
• When simplifying fractions, always be careful to cancel out common factors correctly.
• Don't be afraid to ask for help if you are struggling with a problem.

Final Thoughts

Simplifying exponents and fractions is a fundamental skill in mathematics that can be applied to a wide range of problems. By understanding and applying the properties of exponents and fractions, you can solve complex problems and make informed decisions. Remember to practice regularly and seek help when needed. With time and practice, you will become more comfortable and confident in your ability to simplify exponents and fractions.

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 simplified the exponents and fractions. In this article, we will answer some frequently asked questions about simplifying exponents and fractions.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to a certain exponent. For example, $2^3$ is an exponent, while $8$ is the power.

Q: How do I simplify exponents with different bases?

A: To simplify exponents with different bases, you need to find the least common multiple (LCM) of the bases and then rewrite the exponents in terms of the LCM. For example, $\frac{2^3}{3^2} = \frac{2^3 \cdot 3^{-2}}{3^2 \cdot 3^{-2}} = \frac{2^3 \cdot 3^{-2}}{1} = 2^3 \cdot 3^{-2}$.

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

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

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

A: When dividing exponents with the same base, you subtract the exponents. For example, $\frac{2^3}{2^4} = 2^{3-4} = 2^{-1}$.

Q: How do I simplify fractions with exponents?

A: To simplify fractions with exponents, you need to simplify the numerator and denominator separately and then divide the two simplified expressions. For example, $\frac{2^3}{2^4} = \frac{2^3}{2^4} = 2^{3-4} = 2^{-1}$.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you need to rewrite the expression with a positive exponent. For example, $2^{-3} = \frac{1}{2^3}$.

Q: What is the rule for simplifying expressions with zero exponents?

A: When simplifying expressions with zero exponents, you can ignore