4. Simplify The Following Expressions:a) $\left(4^3\right)^5$b) $\left(6^2\right)^4$

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying two exponential expressions: $\left(4^3\right)^5$ and $\left(6^2\right)^4$. We will use the properties of exponents to simplify these expressions and provide a clear understanding of the underlying concepts.

Property of Exponents

Before we dive into the simplification process, it's essential to understand the property of exponents. The property states that when we raise a power to another power, we multiply the exponents. Mathematically, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the exponent, and $n$ is the power.

Simplifying $\left(4^3\right)^5$

Now that we have a clear understanding of the property of exponents, let's simplify the expression $\left(4^3\right)^5$. Using the property of exponents, we can rewrite the expression as:

$\left(4^3\right)^5 = 4^{3 \cdot 5}$

Next, we multiply the exponents:

$3 \cdot 5 = 15$

So, the expression becomes:

$4^{15}$

This is the simplified form of the expression $\left(4^3\right)^5$.

Simplifying $\left(6^2\right)^4$

Similarly, let's simplify the expression $\left(6^2\right)^4$. Using the property of exponents, we can rewrite the expression as:

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

Next, we multiply the exponents:

$2 \cdot 4 = 8$

So, the expression becomes:

$6^8$

This is the simplified form of the expression $\left(6^2\right)^4$.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill in mathematics. By understanding the property of exponents, we can simplify complex expressions and provide a clear understanding of the underlying concepts. In this article, we simplified two exponential expressions: $\left(4^3\right)^5$ and $\left(6^2\right)^4$. We used the property of exponents to rewrite the expressions and then multiplied the exponents to simplify them. The simplified forms of the expressions are $4^{15}$ and $6^8$, respectively.

Real-World Applications

Simplifying exponential expressions has numerous real-world applications. In finance, for example, exponential expressions are used to calculate compound interest. In science, exponential expressions are used to model population growth and decay. In engineering, exponential expressions are used to design and optimize systems.

Tips and Tricks

When simplifying exponential expressions, it's essential to remember the following tips and tricks:

• Use the property of exponents to rewrite the expression.
• Multiply the exponents to simplify the expression.
• Be careful when multiplying exponents, as the order of operations matters.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

• Simplify the expression $\left(3^4\right)^6$.
• Simplify the expression $\left(2^5\right)^3$.

Answer Key

• $\left(3^4\right)^6 = 3^{4 \cdot 6} = 3^{24}$
• $\left(2^5\right)^3 = 2^{5 \cdot 3} = 2^{15}$

Final Thoughts

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Introduction

In our previous article, we discussed the concept of simplifying exponential expressions using the property of exponents. We simplified two exponential expressions: $\left(4^3\right)^5$ and $\left(6^2\right)^4$. In this article, we will provide a Q&A section to help you better understand the concept and provide additional practice problems.

Q&A

Q: What is the property of exponents?

A: The property of exponents states that when we raise a power to another power, we multiply the exponents. Mathematically, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

Q: How do I simplify an exponential expression using the property of exponents?

A: To simplify an exponential expression using the property of exponents, follow these steps:

1. Rewrite the expression using the property of exponents.
2. Multiply the exponents.
3. Simplify the resulting expression.

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

A: An exponent is a number that is raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3$, 3 is the exponent and 2 is the base.

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

A: Yes, you can simplify an exponential expression with a negative exponent. To do this, follow these steps:

1. Rewrite the expression using the property of exponents.
2. Multiply the exponents.
3. Simplify the resulting expression.

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

A: An exponential expression with a zero exponent is equal to 1. For example, $a^0 = 1$.

Q: Can I simplify an exponential expression with a fractional exponent?

A: Yes, you can simplify an exponential expression with a fractional exponent. To do this, follow these steps:

1. Rewrite the expression using the property of exponents.
2. Multiply the exponents.
3. Simplify the resulting expression.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

• Simplify the expression $\left(5^2\right)^4$.
• Simplify the expression $\left(3^3\right)^2$.
• Simplify the expression $\left(2^4\right)^3$.
• Simplify the expression $\left(4^5\right)^2$.

Answer Key

• $\left(5^2\right)^4 = 5^{2 \cdot 4} = 5^8$
• $\left(3^3\right)^2 = 3^{3 \cdot 2} = 3^6$
• $\left(2^4\right)^3 = 2^{4 \cdot 3} = 2^{12}$
• $\left(4^5\right)^2 = 4^{5 \cdot 2} = 4^{10}$

Final Thoughts

Simplifying exponential expressions is a fundamental skill in mathematics. By understanding the property of exponents and using it to simplify complex expressions, we can provide a clear understanding of the underlying concepts. In this article, we provided a Q&A section to help you better understand the concept and provide additional practice problems. We hope this article has been helpful in your understanding of simplifying exponential expressions.