The Expression \left(2^3\right)\left(2^4\right ] Is Equivalent To:1. 2 7 2^7 2 7 2. 2 12 2^{12} 2 12 3. 4 7 4^7 4 7 4. 4 12 4^{12} 4 12

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Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The expression $\left(2^3\right)\left(2^4\right)$ is a classic example of how exponents can be used to simplify complex expressions. In this article, we will delve into the world of exponents and explore the equivalent forms of the given expression.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." This means that $2^3$ is equal to $2 \times 2 \times 2$, which equals 8.

The Expression $\left(2^3\right)\left(2^4\right)$

Now, let's examine the expression $\left(2^3\right)\left(2^4\right)$. Using the definition of exponents, we can rewrite this expression as:

$$\left(2^3\right)\left(2^4\right) = \left(2 \times 2 \times 2\right)\left(2 \times 2 \times 2 \times 2\right)$$

Simplifying the Expression

To simplify the expression, we can use the associative property of multiplication, which states that the order in which we multiply numbers does not change the result. Therefore, we can rewrite the expression as:

$$\left(2 \times 2 \times 2\right)\left(2 \times 2 \times 2 \times 2\right) = \left(2 \times 2 \times 2 \times 2 \times 2 \times 2\right)$$

Using the Product of Powers Rule

Now, we can use the product of powers rule, which states that when we multiply two numbers with the same base, we can add their exponents. In this case, both numbers have a base of 2. Therefore, we can rewrite the expression as:

$$\left(2 \times 2 \times 2 \times 2 \times 2 \times 2\right) = 2^{3+4}$$

Simplifying the Exponent

Using the definition of exponents, we can simplify the exponent as follows:

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

Conclusion

In conclusion, the expression $\left(2^3\right)\left(2^4\right)$ is equivalent to $2^7$. This is because we can use the associative property of multiplication to simplify the expression, and then use the product of powers rule to add the exponents.

Why is this Important?

Understanding exponents and equivalent forms is crucial in mathematics, as it allows us to simplify complex expressions and solve problems more efficiently. In this article, we have seen how the expression $\left(2^3\right)\left(2^4\right)$ can be simplified using the product of powers rule. This is just one example of how exponents can be used to simplify complex expressions.

Real-World Applications

Exponents have numerous real-world applications, including:

• Finance: Exponents are used to calculate compound interest and investment returns.
• Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
• Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

When working with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to use the product of powers rule: When multiplying two numbers with the same base, make sure to add their exponents.
• Not simplifying the exponent: Make sure to simplify the exponent by adding the exponents of the two numbers.
• Not using the associative property of multiplication: Make sure to use the associative property of multiplication to simplify the expression.

Conclusion

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Introduction

In our previous article, we explored the expression $\left(2^3\right)\left(2^4\right)$ and showed how it can be simplified using the product of powers rule. In this article, we will answer some common questions related to exponents and equivalent forms.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we can add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following steps:

1. Use the associative property of multiplication to simplify the expression.
2. Use the product of powers rule to add the exponents.
3. Simplify the exponent by adding the exponents of the two numbers.

Q: What is the difference between $2^3$ and $2^{3+4}$?

A: $2^3$ is equal to $2 \times 2 \times 2$, which equals 8. $2^{3+4}$ is equal to $2^7$, which equals $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$, which equals 128.

Q: Can I use the product of powers rule with different bases?

A: No, the product of powers rule only applies to numbers with the same base. For example, $2^3 \times 3^4$ cannot be simplified using the product of powers rule.

Q: How do I handle negative exponents?

A: Negative exponents can be handled by using the rule $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: Can I use the product of powers rule with fractions?

A: Yes, the product of powers rule can be used with fractions. For example, $\left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^4 = \left(\frac{1}{2}\right)^{3+4} = \left(\frac{1}{2}\right)^7$.

Q: How do I handle exponents with variables?

A: Exponents with variables can be handled by using the same rules as exponents with numbers. For example, $x^3 \times x^4 = x^{3+4} = x^7$.

Q: Can I use the product of powers rule with exponents with different bases?

A: No, the product of powers rule only applies to numbers with the same base. For example, $2^3 \times 3^4$ cannot be simplified using the product of powers rule.

Conclusion

In conclusion, understanding exponents and equivalent forms is crucial in mathematics. By using the product of powers rule and simplifying exponents, we can solve complex problems more efficiently. We hope that this Q&A article has helped to clarify any questions you may have had about exponents and equivalent forms.

Common Mistakes

When working with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to use the product of powers rule: When multiplying two numbers with the same base, make sure to add their exponents.
• Not simplifying the exponent: Make sure to simplify the exponent by adding the exponents of the two numbers.
• Not using the associative property of multiplication: Make sure to use the associative property of multiplication to simplify the expression.
• Not handling negative exponents correctly: Make sure to use the rule $a^{-n} = \frac{1}{a^n}$ when handling negative exponents.

Real-World Applications

Exponents have numerous real-world applications, including:

• Finance: Exponents are used to calculate compound interest and investment returns.
• Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
• Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, understanding exponents and equivalent forms is crucial in mathematics. By using the product of powers rule and simplifying exponents, we can solve complex problems more efficiently. We hope that this Q&A article has helped to clarify any questions you may have had about exponents and equivalent forms.