Simplify $\left(6^{1 / 4}\right)^4$.A. 6 B. $6^{16}$ C. $6^4$ D. $\frac{1}{6}$

$$

Understanding Exponents and Their Properties

When dealing with exponents, it's essential to understand the properties and rules that govern them. In this case, we're given the expression $\left(6^{1 / 4}\right)^4$ and asked to simplify it. To do this, we need to apply the properties of exponents, specifically the power of a power rule.

The Power of a Power Rule

The power of a power rule states that when we raise a power to another power, we multiply the exponents. In mathematical terms, this can be expressed as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the first exponent, and $n$ is the second exponent.

Applying the Power of a Power Rule

Now that we understand the power of a power rule, let's apply it to the given expression:

$\left(6^{1 / 4}\right)^4$

Using the power of a power rule, we can rewrite this expression as:

$6^{(1 / 4) \cdot 4}$

Simplifying the Expression

Now that we have the expression in the form $6^{m \cdot n}$, we can simplify it by multiplying the exponents:

$6^{(1 / 4) \cdot 4} = 6^{1}$

Evaluating the Final Expression

Now that we have simplified the expression to $6^{1}$, we can evaluate it by applying the definition of an exponent:

$6^{1} = 6$

Therefore, the final answer is:

A. 6

Conclusion

In this article, we applied the power of a power rule to simplify the expression $\left(6^{1 / 4}\right)^4$. By understanding the properties of exponents and applying the power of a power rule, we were able to simplify the expression and arrive at the final answer.

Frequently Asked Questions

• What is the power of a power rule? The power of a power rule states that when we raise a power to another power, we multiply the exponents.
• How do we apply the power of a power rule? To apply the power of a power rule, we multiply the exponents of the two powers.
• What is the final answer to the expression $\left(6^{1 / 4}\right)^4$? The final answer is 6.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Related Articles

• Simplifying Exponential Expressions
• Evaluating Exponential Functions
• Understanding Exponents and Their Properties
  

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying exponential expressions.

Q: What is the power of a power rule?

A: The power of a power rule states that when we raise a power to another power, we multiply the exponents. In mathematical terms, this can be expressed as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the first exponent, and $n$ is the second exponent.

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

A: To apply the power of a power rule, we multiply the exponents of the two powers. For example, if we have the expression $(2^3)^4$, we would multiply the exponents as follows:

$(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$

Q: What is the final answer to the expression $\left(6^{1 / 4}\right)^4$?

A: The final answer to the expression $\left(6^{1 / 4}\right)^4$ is 6. This is because we can apply the power of a power rule to simplify the expression as follows:

$\left(6^{1 / 4}\right)^4 = 6^{(1 / 4) \cdot 4} = 6^{1} = 6$

Q: How do we simplify exponential expressions with negative exponents?

A: To simplify exponential expressions with negative exponents, we can use the rule that states:

$a^{-m} = \frac{1}{a^m}$

For example, if we have the expression $2^{-3}$, we can simplify it as follows:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Q: How do we simplify exponential expressions with fractional exponents?

A: To simplify exponential expressions with fractional exponents, we can use the rule that states:

$a^{m/n} = \sqrt[n]{a^m}$

For example, if we have the expression $2^{3/4}$, we can simplify it as follows:

$2^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8} = 2$

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base raised to a power, such as $2^3$. A polynomial expression is an expression that contains variables raised to various powers, such as $x^2 + 3x - 4$.

Q: How do we evaluate exponential expressions with variables?

A: To evaluate exponential expressions with variables, we can use the rule that states:

$a^m = a \cdot a \cdot a \cdot ... \cdot a$ (m times)

For example, if we have the expression $2^x$, we can evaluate it as follows:

$2^x = 2 \cdot 2 \cdot 2 \cdot ... \cdot 2$ (x times)

Conclusion

In this article, we've answered some of the most frequently asked questions about simplifying exponential expressions. We've covered topics such as the power of a power rule, simplifying exponential expressions with negative exponents, and evaluating exponential expressions with variables.

Frequently Asked Questions (FAQs)

• What is the power of a power rule?
• How do we apply the power of a power rule?
• What is the final answer to the expression $\left(6^{1 / 4}\right)^4$?
• How do we simplify exponential expressions with negative exponents?
• How do we simplify exponential expressions with fractional exponents?
• What is the difference between an exponential expression and a polynomial expression?
• How do we evaluate exponential expressions with variables?

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Related Articles

• Simplifying Exponential Expressions
• Evaluating Exponential Functions
• Understanding Exponents and Their Properties