Simplify The Expression: { \left(2 2\right) 7$}$

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Introduction

In mathematics, expressions involving exponents can be simplified using the rules of exponentiation. One of the most common rules is the power of a power rule, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{m \cdot n}$. In this article, we will use this rule to simplify the expression $\left(2^2\right)^7$.

Understanding the Power of a Power Rule

The power of a power rule is a fundamental concept in mathematics that allows us to simplify expressions involving exponents. This rule states that when we have an exponent raised to another exponent, we can multiply the two exponents together. In other words, $\left(a^m\right)^n = a^{m \cdot n}$. This rule can be applied to any numbers $a$ and $b$ and any integers $m$ and $n$.

Applying the Power of a Power Rule to the Expression

Now that we have a good understanding of the power of a power rule, let's apply it to the expression $\left(2^2\right)^7$. Using the rule, we can simplify the expression as follows:

$\left(2^2\right)^7 = 2^{2 \cdot 7} = 2^{14}$

Simplifying the Expression Further

The expression $2^{14}$ can be simplified further by evaluating the exponent. To do this, we need to multiply the base number $2$ by itself $14$ times. This can be done using the following calculation:

$2^{14} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Evaluating the Exponent

Evaluating the exponent $2^{14}$ involves multiplying the base number $2$ by itself $14$ times. This can be done using the following calculation:

$2^{14} = 16384$

Conclusion

In this article, we used the power of a power rule to simplify the expression $\left(2^2\right)^7$. We applied the rule to the expression and simplified it to $2^{14}$. Finally, we evaluated the exponent $2^{14}$ and found that it equals $16384$. This demonstrates the importance of the power of a power rule in simplifying expressions involving exponents.

Examples and Applications

The power of a power rule has many applications in mathematics and other fields. Here are a few examples:

• Simplifying expressions: The power of a power rule can be used to simplify expressions involving exponents. For example, $\left(3^4\right)^5 = 3^{4 \cdot 5} = 3^{20}$.
• Evaluating exponents: The power of a power rule can be used to evaluate exponents. For example, $2^{14} = 16384$.
• Solving equations: The power of a power rule can be used to solve equations involving exponents. For example, if we have the equation $x^2 = 16$, we can use the power of a power rule to simplify the expression and solve for $x$.

Tips and Tricks

Here are a few tips and tricks for working with the power of a power rule:

• Make sure to apply the rule correctly: When applying the power of a power rule, make sure to multiply the two exponents together.
• Simplify the expression: After applying the power of a power rule, simplify the expression by evaluating the exponent.
• Check your work: Always check your work to make sure that the expression is simplified correctly.

Common Mistakes

Here are a few common mistakes to avoid when working with the power of a power rule:

• Not applying the rule correctly: Make sure to multiply the two exponents together when applying the power of a power rule.
• Not simplifying the expression: After applying the power of a power rule, simplify the expression by evaluating the exponent.
• Not checking your work: Always check your work to make sure that the expression is simplified correctly.

Final Thoughts

The power of a power rule is a fundamental concept in mathematics that allows us to simplify expressions involving exponents. By applying this rule, we can simplify expressions and evaluate exponents. Remember to always apply the rule correctly, simplify the expression, and check your work to ensure that the expression is simplified correctly. With practice and patience, you will become proficient in working with the power of a power rule and be able to simplify expressions involving exponents with ease.

Introduction

In our previous article, we discussed the power of a power rule and how it can be used to simplify expressions involving exponents. In this article, we will answer some of the most frequently asked questions about the power of a power rule and provide additional examples and tips for working with this rule.

Q&A

Q: What is the power of a power rule?

A: The power of a power rule is a fundamental concept in mathematics that allows us to simplify expressions involving exponents. It states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{m \cdot n}$.

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

A: To apply the power of a power rule, simply multiply the two exponents together. For example, $\left(2^2\right)^7 = 2^{2 \cdot 7} = 2^{14}$.

Q: What are some common mistakes to avoid when working with the power of a power rule?

A: Some common mistakes to avoid when working with the power of a power rule include:

• Not applying the rule correctly: Make sure to multiply the two exponents together.
• Not simplifying the expression: After applying the power of a power rule, simplify the expression by evaluating the exponent.
• Not checking your work: Always check your work to make sure that the expression is simplified correctly.

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

A: Yes, you can use the power of a power rule with negative exponents. For example, $\left(2^{-2}\right)^7 = 2^{-2 \cdot 7} = 2^{-14}$.

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

A: Yes, you can use the power of a power rule with fractional exponents. For example, $\left(2^{1/2}\right)^7 = 2^{1/2 \cdot 7} = 2^{7/2}$.

Q: How do I evaluate an expression with a power of a power?

A: To evaluate an expression with a power of a power, simply multiply the two exponents together and then evaluate the resulting expression. For example, $2^{14} = 16384$.

Q: Can I use the power of a power rule to simplify expressions with multiple exponents?

A: Yes, you can use the power of a power rule to simplify expressions with multiple exponents. For example, $\left(2^2\right)^3 \cdot \left(2^4\right)^2 = 2^{2 \cdot 3} \cdot 2^{4 \cdot 2} = 2^6 \cdot 2^8 = 2^{6 + 8} = 2^{14}$.

Examples and Applications

Here are a few more examples of how the power of a power rule can be used to simplify expressions:

• Simplifying expressions: $\left(3^4\right)^5 = 3^{4 \cdot 5} = 3^{20}$
• Evaluating exponents: $2^{14} = 16384$
• Solving equations: If we have the $x^2 = 16$, we can use the power of a power rule to simplify the expression and solve for $x$.

Tips and Tricks

Here are a few more tips and tricks for working with the power of a power rule:

• Make sure to apply the rule correctly: When applying the power of a power rule, make sure to multiply the two exponents together.
• Simplify the expression: After applying the power of a power rule, simplify the expression by evaluating the exponent.
• Check your work: Always check your work to make sure that the expression is simplified correctly.

Conclusion

The power of a power rule is a fundamental concept in mathematics that allows us to simplify expressions involving exponents. By applying this rule, we can simplify expressions and evaluate exponents. Remember to always apply the rule correctly, simplify the expression, and check your work to ensure that the expression is simplified correctly. With practice and patience, you will become proficient in working with the power of a power rule and be able to simplify expressions involving exponents with ease.