Simplify: ( 9 4 ) ( 3 ) 4 \left(9^4\right)(3)^4 ( 9 4 ) ( 3 ) 4

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Introduction

When dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore how to simplify the expression $\left(9^4\right)(3)^4$ using the properties of exponents. We will delve into the world of exponent rules, including the product of powers property, and demonstrate how to apply these rules to simplify the given expression.

Understanding Exponents

Before we dive into simplifying the expression, let's take a moment to understand what exponents are and how they work. An exponent is a small number that is placed above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, in the expression $a^b$, the base is $a$ and the exponent is $b$. This means that $a$ is multiplied by itself $b$ times.

For instance, $2^3$ means that $2$ is multiplied by itself $3$ times, which equals $2 \times 2 \times 2 = 8$. Similarly, $3^4$ means that $3$ is multiplied by itself $4$ times, which equals $3 \times 3 \times 3 \times 3 = 81$.

The Product of Powers Property

One of the most important properties of exponents is the product of powers property. This property states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have the expression $a^m \times a^n$, we can simplify it to $a^{m+n}$.

For example, let's consider the expression $2^3 \times 2^4$. Using the product of powers property, we can simplify this expression to $2^{3+4} = 2^7$. This means that $2^3 \times 2^4 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$.

Simplifying the Expression

Now that we have a solid understanding of exponents and the product of powers property, let's apply these concepts to simplify the expression $\left(9^4\right)(3)^4$.

First, let's rewrite the expression using the fact that $9 = 3^2$. This gives us $\left((3^2)^4\right)(3)^4$.

Next, let's apply the power of a power property, which states that $(a^m)^n = a^{mn}$. Using this property, we can simplify the expression to $3^{2 \times 4} \times 3^4$.

Now, let's apply the product of powers property, which states that $a^m \times a^n = a^{m+n}$. Using this property, we can simplify the expression to $3^{2 \times 4 + 4} = 3^{12}$.

Conclusion

In this article, we have explored how to simplify the expression $\left(9^4\right)(3)^4$ using the properties of exponents. We have applied the product of powers property and the power of a power property to simplify the expression and arrive at the final answer of $3^{12}$.

By understanding the rules and properties of exponents, we can simplify complex expressions and arrive at the correct solution. Whether you're a student, a teacher, or simply someone who loves math, this article has provided you with the tools and knowledge you need to tackle even the most challenging exponent problems.

Frequently Asked Questions

• What is the product of powers property? The product of powers property states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have the expression $a^m \times a^n$, we can simplify it to $a^{m+n}$.
• How do I apply the product of powers property? To apply the product of powers property, simply add the exponents of the two numbers with the same base. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• What is the power of a power property? The power of a power property states that $(a^m)^n = a^{mn}$. This means that when we raise a power to another power, we can multiply the exponents.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Answer

The final answer is: $\boxed{3^{12}}$

Introduction

Exponents can be a challenging topic for many students and math enthusiasts. However, with a solid understanding of the rules and properties of exponents, you can simplify complex expressions and arrive at the correct solution. In this article, we will answer some of the most frequently asked questions about exponents, providing you with a deeper understanding of this important math concept.

Q&A: Exponents

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two numbers with the same base, we can add their exponents. In other words, if we have the expression $a^m \times a^n$, we can simplify it to $a^{m+n}$.

Q: How do I apply the product of powers property?

A: To apply the product of powers property, simply add the exponents of the two numbers with the same base. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: What is the power of a power property?

A: The power of a power property states that $(a^m)^n = a^{mn}$. This means that when we raise a power to another power, we can multiply the exponents.

Q: How do I apply the power of a power property?

A: To apply the power of a power property, simply multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of zero is equal to 1. In other words, $a^0 = 1$ for any non-zero number $a$.

Q: How do I apply the zero exponent rule?

A: To apply the zero exponent rule, simply replace the exponent with 1. For example, $2^0 = 1$.

Q: What is the negative exponent rule?

A: The negative exponent rule states that any number raised to a negative power is equal to the reciprocal of the number raised to the positive power. In other words, $a^{-n} = \frac{1}{a^n}$.

Q: How do I apply the negative exponent rule?

A: To apply the negative exponent rule, simply replace the negative exponent with a positive exponent and take the reciprocal of the number. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the rule for multiplying exponential expressions with different bases?

A: When multiplying exponential expressions with different bases, we cannot simply add the exponents. Instead, we must multiply the bases and keep the exponents the same. For example, $2^3 \times 3^4 = (2 \times 3)^3 \times 3^4 = 6^3 \times 3^4$.

Q: How do I apply the rule for multiplying exponential expressions with different bases?

A: To apply the rule for multiplying exponential expressions with different bases, simply multiply the bases and keep the exponents the same. For example, $2^3 \times 3^4 = (2 \times 3)^3 \times 3^4 = 6^3 \times 3^4$.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponents, providing you with a deeper understanding of this important math concept. Whether you're a student, a teacher, or simply someone who loves math, this article has provided you with the tools and knowledge you need to tackle even the most challenging exponent problems.

Frequently Asked Questions

• What is the product of powers property?
• How do I apply the product of powers property?
• What is the power of a power property?
• How do I apply the power of a power property?
• What is the zero exponent rule?
• How do I apply the zero exponent rule?
• What is the negative exponent rule?
• How do I apply the negative exponent rule?
• What is the rule for multiplying exponential expressions with different bases?
• How do I apply the rule for multiplying exponential expressions with different bases?

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Answer

The final answer is: Exponents are a fundamental concept in mathematics, and understanding the rules and properties of exponents is essential for simplifying complex expressions and arriving at the correct solution.