Simplify The Expression: $\left(\left(\frac{2}{3}\right)^4\right)^6$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common techniques used to simplify expressions is exponentiation. In this article, we will focus on simplifying the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$ using the properties of exponents.

Understanding Exponents

Before we dive into simplifying the expression, let's quickly review the basics of exponents. An exponent is a small number that is written to the upper right of a number or a variable. It represents the number of times the base is multiplied by itself. For example, $a^b$ means $a$ is multiplied by itself $b$ times.

Properties of Exponents

There are several properties of exponents that we will use to simplify the expression. These properties include:

• Product of Powers: When we multiply two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Simplifying the Expression

Now that we have reviewed the basics of exponents and their properties, let's simplify the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$. We can start by applying the Power of a Power property, which states that $(a^m)^n = a^{m \cdot n}$.

$\left(\left(\frac{2}{3}\right)^4\right)^6 = \left(\frac{2}{3}\right)^{4 \cdot 6}$

Using the Product of Powers property, we can simplify the exponent $4 \cdot 6$ to $24$.

$\left(\frac{2}{3}\right)^{4 \cdot 6} = \left(\frac{2}{3}\right)^{24}$

Now, we can simplify the expression further by applying the Product of Powers property again. We can rewrite the expression as $\frac{2^{24}}{3^{24}}$.

$\left(\frac{2}{3}\right)^{24} = \frac{2^{24}}{3^{24}}$

Conclusion

In this article, we simplified the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$ using the properties of exponents. We applied the Power of a Power and Product of Powers properties to simplify the expression and arrived at the final answer of $\frac{2^{24}}{3^{24}}$. This example demonstrates the importance of understanding the properties of exponents in simplifying expressions and solving mathematical problems.

Frequently Asked Questions

• What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$?
  • The property of exponents that states $(a^m)^n = a^{m \cdot n}$ is called the Power of a Power property.
• How do we simplify the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$?
  • We can simplify the expression by applying the Power of a Power and Product of Powers properties.
• What is the final answer to the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$?
  • The final answer to the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$ is $\frac{2^{24}}{3^{24}}$.

Further Reading

• Exponents and Powers: This article provides a comprehensive overview of exponents and powers, including their properties and applications.
• Simplifying Expressions: This article provides tips and techniques for simplifying expressions, including the use of exponents and powers.
• Mathematical Properties: This article provides an overview of mathematical properties, including the commutative, associative, and distributive properties.
  

Introduction

In our previous article, we simplified the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$ using the properties of exponents. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q&A

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

A: The property of exponents that states $(a^m)^n = a^{m \cdot n}$ is called the Power of a Power property.

Q: How do we simplify the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$?

A: We can simplify the expression by applying the Power of a Power and Product of Powers properties.

Q: What is the final answer to the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$?

A: The final answer to the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$ is $\frac{2^{24}}{3^{24}}$.

Q: What is the difference between the Power of a Power and Product of Powers properties?

A: The Power of a Power property states that $(a^m)^n = a^{m \cdot n}$, while the Product of Powers property states that $a^m \cdot a^n = a^{m+n}$.

Q: Can we simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents by using the property $a^{-n} = \frac{1}{a^n}$.

Q: How do we simplify expressions with fractional exponents?

A: We can simplify expressions with fractional exponents by using the property $a^{m/n} = \sqrt[n]{a^m}$.

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

A: When simplifying expressions with exponents and fractions, we can use the property $\frac{a^m}{a^n} = a^{m-n}$.

Examples

Example 1: Simplifying an Expression with a Power of a Power

Simplify the expression $\left(\left(\frac{2}{3}\right)^4\right)^6$.

$\left(\left(\frac{2}{3}\right)^4\right)^6 = \left(\frac{2}{3}\right)^{4 \cdot 6}$
$\left(\frac{2}{3}\right)^{4 \cdot 6} = \left(\frac{2}{3}\right)^{24}$
$\left(\frac{2}{3}\right)^{24} = \frac{2^{24}}{3^{24}}$

Example 2: Simplifying an Expression with a Product of Powers

Simplify the expression $\frac{2^4 \cdot 2^6}{3^4 \cdot 3^6}$.

$\frac{2^4 \cdot 2^6}{3^4 \cdot 3^6} = \frac{2^{4+6}}{3^{4+6}}$
$\frac{2^{4+6}}{3^{4+6}} = \frac{2^{10}}{3^{10}}$

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with exponents. We also provided examples of how to simplify expressions with powers of a power and products of powers. By understanding the properties of exponents and how to apply them, we can simplify complex expressions and solve mathematical problems with ease.

Further Reading

• Exponents and Powers: This article provides a comprehensive overview of exponents and powers, including their properties and applications.
• Simplifying Expressions: This article provides tips and techniques for simplifying expressions, including the use of exponents and powers.
• Mathematical Properties: This article provides an overview of mathematical properties, including the commutative, associative, and distributive properties.