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

Introduction

Exponents and indices are fundamental concepts in mathematics that play a crucial role in simplifying complex expressions. In this article, we will delve into the world of exponents and indices, focusing on simplifying the given expression: $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$. We will explore the properties of exponents, indices, and their applications in simplifying expressions.

Understanding Exponents and Indices

Exponents and indices are used to represent repeated multiplication of a number. The exponent is the small number that is written above and to the right of the base number, indicating how many times the base number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. The index, on the other hand, is the power to which the base number is raised. In the expression $2^3$, the index is $3$.

Properties of Exponents and Indices

There are several properties of exponents and indices that are essential to understand when simplifying expressions. These properties include:

• Product of Powers: When multiplying two powers with the same base, add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.
• Power of a Power: When raising a power to another power, multiply the exponents. For example, $(2^3)^4 = 2^{3\cdot4} = 2^{12}$.
• Quotient of Powers: When dividing two powers with the same base, subtract the exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to $1$. For example, $2^0 = 1$.

Simplifying the Expression

Now that we have a solid understanding of exponents and indices, let's apply these concepts to simplify the given expression: $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$.

Step 1: Simplify the Numerator

Using the Product of Powers property, we can simplify the numerator by adding the exponents:

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

Step 2: Simplify the Denominator

Using the Power of a Power property, we can simplify the denominator by multiplying the exponents:

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

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator:

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

Conclusion

In this article, we have explored the properties of exponents and indices and applied these concepts to simplify the given expression: $\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}$. We have demonstrated the importance of understanding exponents and indices in simplifying complex expressions. By applying the properties of exponents and indices, we have arrived at the simplified expression: $1$. This article has provided a comprehensive guide to exponents and indices, making it an essential resource for anyone looking to improve their mathematical skills.

Frequently Asked Questions

• What is the difference between an exponent and an index? An exponent is the small number that is written above and to the right of the base number, indicating how many times the base number should be multiplied by itself. An index, on the other hand, is the power to which the base number is raised.
• How do I simplify an expression with exponents and indices? To simplify an expression with exponents and indices, apply the properties of exponents and indices, such as the product of powers, power of a power, quotient of powers, and zero exponent.
• What is the value of $2^0$? The value of $2^0$ is $1$, as any non-zero number raised to the power of zero is equal to $1$.

Further Reading

• Exponents and Indices: A Comprehensive Guide This article provides a detailed explanation of exponents and indices, including their properties and applications.
• Simplifying Expressions with Exponents and Indices This article provides a step-by-step guide to simplifying expressions with exponents and indices.
• Mathematics: A Subject of Beauty and Wonder This article explores the beauty and wonder of mathematics, including its applications and importance in our daily lives.
  

Introduction

Exponents and indices are fundamental concepts in mathematics that play a crucial role in simplifying complex expressions. In this article, we will address some of the most frequently asked questions about exponents and indices, providing a comprehensive guide to help you better understand these concepts.

Q&A

Q: What is the difference between an exponent and an index?

A: An exponent is the small number that is written above and to the right of the base number, indicating how many times the base number should be multiplied by itself. An index, on the other hand, is the power to which the base number is raised.

Q: How do I simplify an expression with exponents and indices?

A: To simplify an expression with exponents and indices, apply the properties of exponents and indices, such as the product of powers, power of a power, quotient of powers, and zero exponent.

Q: What is the value of $2^0$?

A: The value of $2^0$ is $1$, as any non-zero number raised to the power of zero is equal to $1$.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by using the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

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

A: To simplify an expression with a fractional exponent, apply the property of fractional exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by using the property of zero exponents, which states that $a^0 = 1$, where $a$ is a non-zero number.

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

A: To simplify an expression with a negative base, apply the property of negative bases, which states that $(-a)^n = a^n$ if $n$ is even, and $(-a)^n = -a^n$ if $n$ is odd.

Q: Can I simplify an expression with a complex number as the base?

A: Yes, you can simplify an expression with a complex number as the base by using the properties of complex numbers, such as the fact that $i^2 = -1$.

Examples

Example 1: Simplifying an Expression with Exponents and Indices

Simplify the expression: $\frac{2^3 \cdot 2^4}{2^2}$.

Solution:

Using the property of exponents, we can simplify the numerator by adding the exponents:

$2^3 \cdot 2^4 = 2^{3+4} = 2^7$

Using the property of exponents, we can simplify the denominator by subtracting the exponents:

$2^2 = 2^{2-2} = 2^0 = 1$

Therefore, the expression simplifies to:

$\frac{2^7}{1} = 2^7 = 128$

Example 2: Simplifying an Expression with a Negative Exponent

Simplify the expression: $2^{-3}$.

Solution:

Using the property of negative exponents, we can rewrite the expression as:

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

Conclusion

In this article, we have addressed some of the most frequently asked questions about exponents and indices, providing a comprehensive guide to help you better understand these concepts. By applying the properties of exponents and indices, you can simplify complex expressions and solve a wide range of mathematical problems.

Further Reading

• Exponents and Indices: A Comprehensive Guide This article provides a detailed explanation of exponents and indices, including their properties and applications.
• Simplifying Expressions with Exponents and Indices This article provides a step-by-step guide to simplifying expressions with exponents and indices.
• Mathematics: A Subject of Beauty and Wonder This article explores the beauty and wonder of mathematics, including its applications and importance in our daily lives.

Frequently Asked Questions

• What is the difference between an exponent and an index? An exponent is the small number that is written above and to the right of the base number, indicating how many times the base number should be multiplied by itself. An index, on the other hand, is the power to which the base number is raised.
• How do I simplify an expression with exponents and indices? To simplify an expression with exponents and indices, apply the properties of exponents and indices, such as the product of powers, power of a power, quotient of powers, and zero exponent.
• What is the value of $2^0$? The value of $2^0$ is $1$, as any non-zero number raised to the power of zero is equal to $1$.