Applying the Power of a Power Rule to the Given Expression

Oct 13, 2025 by windailys.com 46 views

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules and properties that govern them. One of the key properties of exponents is the power of a power rule, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$ , $(a^m)^n = a^{m \cdot n}$ . In this article, we will explore this rule and its application to the given expression $\left(2^9\right)^8$ .

The Power of a Power Rule

The power of a power rule is a fundamental concept in mathematics that helps us simplify expressions involving exponents. This rule states that when we have an exponent raised to another exponent, we can multiply the exponents together. In other words, $(a^m)^n = a^{m \cdot n}$ . 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 Given Expression

Now that we have a good understanding of the power of a power rule, let's apply it to the given expression $\left(2^9\right)^8$ . Using the power of a power rule, we can simplify this expression by multiplying the exponents together. Therefore, $\left(2^9\right)^8 = 2^{9 \cdot 8} = 2^{72}$ .

Evaluating the Equivalent Expressions

Now that we have simplified the given expression to $2^{72}$ , let's evaluate the equivalent expressions provided in the options. We will compare each option with the simplified expression to determine which one is equivalent.

Option A: $\frac{1}{2^{17}}$

To evaluate this option, we need to simplify the expression. Using the rule of negative exponents, we can rewrite $\frac{1}{2^{17}}$ as $2^{-17}$ . However, this is not equivalent to the simplified expression $2^{72}$ .

Option B: $\frac{1}{2^{-72}}$

To evaluate this option, we need to simplify the expression. Using the rule of negative exponents, we can rewrite $\frac{1}{2^{-72}}$ as $2^{72}$ . This option is equivalent to the simplified expression.

Option C: $\frac{1}{2^{72}}$

To evaluate this option, we need to simplify the expression. Using the rule of negative exponents, we can rewrite $\frac{1}{2^{72}}$ as $2^{-72}$ . However, this is not equivalent to the simplified expression $2^{72}$ .

Option D: $2^{17}$

To evaluate this option, we need to simplify the expression. However, this option is not equivalent to the simplified expression $2^{72}$ .

Conclusion

In conclusion, the expression $\left(2^9\right)^8$ is equivalent to $2^{72}$ . Among the given options, only option B, $\frac{1}{2^{-72}}$ , is equivalent to the simplified expression. This is because option B can be rewritten as $2^{72}$ using the rule of negative exponents.

Key Takeaways

The power of a power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$ , $(a^m)^n = a^{m \cdot n}$ .
The power of a power rule can be applied to any numbers $a$ and $b$ and any integers $m$ and $n$ .
The rule of negative exponents states that for any number $a$ and any integer $n$ , $a^{-n} = \frac{1}{a^n}$ .
The expression $\left(2^9\right)^8$ is equivalent to $2^{72}$ .
Among the given options, only option B, $\frac{1}{2^{-72}}$ , is equivalent to the simplified expression.
Frequently Asked Questions (FAQs) =====================================

In this article, we will address some of the most frequently asked questions related to the power of a power rule and equivalent expressions.

Q: What is the power of a power rule?

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

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

A: To apply the power of a power rule, simply multiply the exponents together. For example, if we have the expression $(2^3)^4$ , we can simplify it by multiplying the exponents together: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ .

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that for any number $a$ and any integer $n$ , $a^{-n} = \frac{1}{a^n}$ . This means that a negative exponent can be rewritten as a fraction with the number in the denominator.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, simply rewrite the negative exponent as a fraction with the number in the denominator. For example, if we have the expression $2^{-3}$ , we can simplify it by rewriting the negative exponent as a fraction: $2^{-3} = \frac{1}{2^3}$ .

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents repeated division of a number. For example, $2^3$ represents $2 \cdot 2 \cdot 2$ , while $2^{-3}$ represents $\frac{1}{2 \cdot 2 \cdot 2}$ .

Q: How do I evaluate expressions with exponents?

A: To evaluate expressions with exponents, simply apply the power of a power rule and the rule of negative exponents as needed. For example, if we have the expression $(2^3)^4$ , we can simplify it by multiplying the exponents together: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ .

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

Forgetting to multiply the exponents together when applying the power of a power rule
Not rewriting negative exponents as fractions
Not simplifying expressions with exponents correctly

Q: How do I practice working with exponents?

A: To practice working with exponents, try the following:

Simplify expressions with exponents using the power of a power rule and the rule of negative exponents
Evaluate expressions with exponents
Practice applying the power of a power rule and the rule of negative exponents to different types of expressions

Conclusion

In conclusion, the power of a power rule and equivalent expressions are fundamental concepts in mathematics that help us simplify expressions involving exponents. By understanding and applying these concepts, we can evaluate expressions with exponents correctly and avoid common mistakes.