Examine And Complete The Work Simplifying The Expression ( 2 3 ) 3 \left(2^3\right)^3 ( 2 3 ) 3 .1. Expand Using 3 Factors Of 2 3 2^3 2 3 : \left(2^3\right)\left(2^3\right)\left(2^3\right ]2. Apply The Product Of A Power: $2^{3+3+3}$3.

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 examine and complete the work of simplifying the expression $\left(2^3\right)^3$ using two different methods: expanding using three factors of $2^3$ and applying the product of a power.

Method 1: Expanding Using Three Factors of $2^3$

To simplify the expression $\left(2^3\right)^3$, we can start by expanding it using three factors of $2^3$. This means that we will multiply the expression by itself three times.

$\left(2^3\right)^3 = \left(2^3\right)\left(2^3\right)\left(2^3\right)$

Now, let's expand each factor:

$\left(2^3\right)\left(2^3\right)\left(2^3\right) = \left(2 \cdot 2 \cdot 2\right)\left(2 \cdot 2 \cdot 2\right)\left(2 \cdot 2 \cdot 2\right)$

Simplifying each factor, we get:

$\left(2 \cdot 2 \cdot 2\right)\left(2 \cdot 2 \cdot 2\right)\left(2 \cdot 2 \cdot 2\right) = 8 \cdot 8 \cdot 8$

Now, let's multiply the numbers:

$8 \cdot 8 \cdot 8 = 512$

Therefore, the simplified expression is $2^{3+3+3}$.

Method 2: Applying the Product of a Power

Another way to simplify the expression $\left(2^3\right)^3$ is by applying the product of a power. This rule states that when we multiply two powers with the same base, we add their exponents.

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

Using the product of a power rule, we can simplify the expression as follows:

$2^{3+3+3} = 2^9$

Therefore, the simplified expression is $2^9$.

Discussion

In this article, we have examined and completed the work of simplifying the expression $\left(2^3\right)^3$ using two different methods: expanding using three factors of $2^3$ and applying the product of a power. Both methods have led to the same simplified expression, $2^9$.

The product of a power rule is a powerful tool for simplifying exponential expressions. It allows us to combine powers with the same base and add their exponents. This rule is essential for solving problems involving exponential expressions and is a fundamental concept in algebra.

Conclusion

In conclusion, simplifying exponential expressions is an essential skill for students and professionals alike. By using the product of a power rule, we can simplify expressions like $\left(2^3\right)^3$ and arrive at the same result as expanding using three factors of $2^3$. This rule is a fundamental concept in algebra and is used extensively in mathematics and science.

Examples and Exercises

1. Simplify the expression $\left(3^2\right)^4$ using the product of a power rule.
2. Simplify the expression $\left(4^3\right)^2$ using the product of a power rule.
3. Simplify the expression $\left(5^4\right)^3$ using the product of a power rule.

Answer Key

1. $3^{2+4} = 3^6$
2. $4^{3+2} = 4^5$
3. $5^{4+3} = 5^7$

Glossary

• Exponential expression: An expression that involves a base raised to a power, such as $2^3$.
• Product of a power: A rule that states when we multiply two powers with the same base, we add their exponents.
• Simplifying exponential expressions: The process of rewriting an exponential expression in a simpler form, such as $2^9$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  Q&A: Simplifying Exponential Expressions =============================================

Frequently Asked Questions

Q: What is an exponential expression?

A: An exponential expression is an expression that involves a base raised to a power, such as $2^3$.

Q: What is the product of a power rule?

A: The product of a power rule is a rule that states when we multiply two powers with the same base, we add their exponents.

Q: How do I simplify an exponential expression using the product of a power rule?

A: To simplify an exponential expression using the product of a power rule, you need to multiply the exponents of the two powers and keep the same base.

Q: Can I simplify an exponential expression using the product of a power rule if the bases are different?

A: No, the product of a power rule only applies when the bases are the same. If the bases are different, you cannot simplify the expression using this rule.

Q: What is the difference between expanding using three factors and applying the product of a power?

A: Expanding using three factors involves multiplying the expression by itself three times, while applying the product of a power involves adding the exponents of the two powers.

Q: Can I simplify an exponential expression using both expanding and applying the product of a power?

A: Yes, you can simplify an exponential expression using both expanding and applying the product of a power. However, the result will be the same.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Not following the order of operations
• Not using the product of a power rule correctly
• Not simplifying the expression completely

Q: How do I know when to use the product of a power rule?

A: You should use the product of a power rule when you have an exponential expression that involves multiplying two powers with the same base.

Q: Can I use the product of a power rule with negative exponents?

A: Yes, you can use the product of a power rule with negative exponents. However, you need to be careful when simplifying the expression.

Q: What are some real-world applications of simplifying exponential expressions?

A: Simplifying exponential expressions has many real-world applications, including:

• Calculating compound interest
• Modeling population growth
• Analyzing data in science and engineering

Q: How do I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through examples and exercises, such as those found in algebra textbooks or online resources.

Additional Resources

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Glossary

• Exponential expression: An expression that involves a base raised to a power, such as $2^3$.
• Product of a power: A rule that states when we multiply two powers with the same base, we add their exponents.
• Simplifying exponential expressions: The process of rewriting an exponential expression in a simpler form, such as $2^9$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer