Understanding Exponents and Power Raised to a Power

Oct 13, 2025 by magicalceylontours.com 31 views

Which Power Raised to a Power Expressions are Equivalent to $8^{12}$ ?

Understanding Exponents and Power Raised to a Power

In mathematics, exponents and power raised to a power are fundamental concepts that help us simplify complex expressions. When we have an expression in the form of $a^m$ , where $a$ is the base and $m$ is the exponent, we can raise it to another power, resulting in $a^{m \cdot n}$ . This concept is crucial in solving various mathematical problems, including those involving exponential expressions.

The Concept of Power Raised to a Power

When we have an expression in the form of $(a^m)^n$ , we can simplify it by multiplying the exponents, resulting in $a^{m \cdot n}$ . This is a fundamental property of exponents that helps us simplify complex expressions. For example, if we have the expression $(2^3)^4$ , we can simplify it by multiplying the exponents, resulting in $2^{3 \cdot 4} = 2^{12}$ .

Evaluating the Given Expressions

Now, let's evaluate the given expressions to determine which ones are equivalent to $8^{12}$ .

$\left(8^{-3}\right)^{-4}$

To evaluate this expression, we need to apply the property of power raised to a power. We can simplify it by multiplying the exponents, resulting in $8^{-3 \cdot -4} = 8^{12}$ .

$\left(8^{-2}\right)^6$

To evaluate this expression, we need to apply the property of power raised to a power. We can simplify it by multiplying the exponents, resulting in $8^{-2 \cdot 6} = 8^{-12}$ . However, this expression is not equivalent to $8^{12}$ .

$\left(8^0\right)^{12}$

To evaluate this expression, we need to apply the property of power raised to a power. We can simplify it by multiplying the exponents, resulting in $8^{0 \cdot 12} = 8^0$ . However, this expression is not equivalent to $8^{12}$ .

$\left(8^1\right)^{12}$

To evaluate this expression, we need to apply the property of power raised to a power. We can simplify it by multiplying the exponents, resulting in $8^{1 \cdot 12} = 8^{12}$ .

Conclusion

Based on the evaluation of the given expressions, we can conclude that the following expressions are equivalent to $8^{12}$ :

$\left(8^{-3}\right)^{-4}$
$\left(8^1\right)^{12}$

These expressions can be simplified by applying the property of power raised to a power, resulting in $8^{12}$ . The other expressions, $\left(8^{-2}\right)^6$ and $\left(8^0\right)^{12}$ , are not equivalent to $8^{12}$ .

Key Takeaways

Exponents and power raised to a power are fundamental concepts in mathematics.
The property of power raised to a power states that $(a^m)^n = a^{m \cdot n}$ .
We can simplify complex expressions by applying this property.
The expressions $\left(8^{-3}\right)^{-4}$ and $\left(8^1\right)^{12}$ are equivalent to $8^{12}$ .

Common Mistakes to Avoid

Not applying the property of power raised to a power when simplifying complex expressions.
Not multiplying the exponents correctly when applying the property of power raised to a power.

Real-World Applications

Exponents and power raised to a power are used in various real-world applications, including finance, science, and engineering.
Understanding these concepts is crucial in solving complex problems in these fields.

Practice Problems

Simplify the expression $(3^4)^5$ .
Simplify the expression $(2^3)^{-2}$ .
Simplify the expression $(5^0)^{10}$ .

Conclusion

In conclusion, understanding exponents and power raised to a power is crucial in simplifying complex expressions. By applying the property of power raised to a power, we can simplify expressions and solve complex problems. The expressions $\left(8^{-3}\right)^{-4}$ and $\left(8^1\right)^{12}$ are equivalent to $8^{12}$ , and we can use this knowledge to solve various mathematical problems.
Q&A: Exponents and Power Raised to a Power

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponents and power raised to a power.

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

A: An exponent is a number that is raised to a power, while a power raised to a power is an expression in the form of $(a^m)^n$ .

Q: How do I simplify an expression with a power raised to a power?

A: To simplify an expression with a power raised to a power, you can multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ .

Q: What is the property of power raised to a power?

A: The property of power raised to a power states that $(a^m)^n = a^{m \cdot n}$ .

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

A: Yes, you can simplify an expression with a negative exponent by applying the property of power raised to a power. For example, $(2^{-3})^4 = 2^{-3 \cdot 4} = 2^{-12}$ .

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the property of power raised to a power. For example, $(2^0)^4 = 2^{0 \cdot 4} = 2^0 = 1$ .

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

A: Yes, you can simplify an expression with a fractional exponent by applying the property of power raised to a power. For example, $(2^{1/2})^4 = 2^{(1/2) \cdot 4} = 2^2 = 4$ .

Q: What are some common mistakes to avoid when simplifying expressions with power raised to a power?

A: Some common mistakes to avoid when simplifying expressions with power raised to a power include:

Not applying the property of power raised to a power
Not multiplying the exponents correctly
Not simplifying the expression correctly

Q: How do I apply the property of power raised to a power in real-world applications?

A: The property of power raised to a power is used in various real-world applications, including finance, science, and engineering. For example, in finance, you may need to calculate the future value of an investment using the formula $FV = PV \cdot (1 + r)^n$ , where $FV$ is the future value, $PV$ is the present value, $r$ is the interest rate, and $n$ is the number of periods.

Q: Can I use a calculator to simplify expressions with power raised to a power?

A: Yes, you can use a calculator to simplify expressions with power raised to a power. However, it's always a good idea to double-check your work by applying the property of power raised to a power manually.

Conclusion

In conclusion, understanding exponents and power raised to a power is crucial in simplifying complex expressions. By applying the property of power raised to a power, you can simplify expressions and solve complex problems. We hope this Q&A article has helped you understand exponents and power raised to a power better.

Practice Problems

Simplify the expression $(3^4)^5$ .
Simplify the expression $(2^3)^{-2}$ .
Simplify the expression $(5^0)^{10}$ .

Additional Resources

Khan Academy: Exponents and Power Raised to a Power
Mathway: Exponents and Power Raised to a Power
Wolfram Alpha: Exponents and Power Raised to a Power

Conclusion

We hope this Q&A article has helped you understand exponents and power raised to a power better. If you have any further questions or need additional help, please don't hesitate to ask.