If $11 A=\sqrt[6]{11 5}$, What Is The Value Of $a$?

Mar 8, 2025 by ADMIN 56 views

$If $11^a=\sqrt[6]{11^5}$, what is the value of $a$?$

Understanding the Problem

The given equation is $11^{a=\sqrt[6]{11}5}$. To find the value of $a$, we need to simplify the expression on the right-hand side and then equate the exponents.

Simplifying the Right-Hand Side

The expression $\sqrt[6]{11^5}$ can be simplified using the rule $\sqrt[n]{x^n}=x$. Applying this rule, we get:

\sqrt[6]{11^5} = (11^5)^{\frac{1}{6}}

Applying the Power of a Power Rule

Using the power of a power rule, which states that $(x^m)n=x^{mn}$, we can simplify the expression further:

(11^5)^{\frac{1}{6}} = 11^{5\cdot\frac{1}{6}}

Simplifying the Exponent

Now, we can simplify the exponent $5\cdot\frac{1}{6}$:

5\cdot\frac{1}{6} = \frac{5}{6}

Equating the Exponents

Now that we have simplified the right-hand side, we can equate the exponents:

11^a = 11^{\frac{5}{6}}

Equating the Exponents (continued)

Since the bases are the same, we can equate the exponents:

a = \frac{5}{6}

Conclusion

Therefore, the value of $a$ is $\frac{5}{6}$.

Example Use Case

This problem can be used to demonstrate the concept of exponent rules and how to simplify expressions with exponents.

Step-by-Step Solution

Simplify the right-hand side of the equation using the rule $\sqrt[n]{x^n}=x$.
Apply the power of a power rule to simplify the expression further.
Simplify the exponent using the rule $m\cdot\frac{n}{p}=\frac{mn}{p}$.
Equate the exponents and solve for $a$.

Common Mistakes

Failing to simplify the right-hand side of the equation.
Not applying the power of a power rule correctly.
Not simplifying the exponent correctly.

Real-World Applications

This problem can be used to demonstrate the concept of exponent rules and how to simplify expressions with exponents in real-world applications, such as finance, science, and engineering.

Q: What is the given equation?

A: The given equation is $11^{a=\sqrt[6]{11}5}$.

Q: How do we simplify the right-hand side of the equation?

A: We can simplify the right-hand side using the rule $\sqrt[n]{x^n}=x$.

Q: What is the simplified form of the right-hand side?

A: The simplified form of the right-hand side is $(11⁵⁾{\frac{1}{6}}$.

Q: How do we simplify the expression further?

A: We can simplify the expression further using the power of a power rule, which states that $(x^m)n=x^{mn}$.

Q: What is the simplified form of the expression?

A: The simplified form of the expression is $11^{5\cdot\frac{1}{6}}$.

Q: How do we simplify the exponent?

A: We can simplify the exponent by multiplying $5$ and $\frac{1}{6}$.

Q: What is the simplified form of the exponent?

A: The simplified form of the exponent is $\frac{5}{6}$.

Q: How do we equate the exponents?

A: Since the bases are the same, we can equate the exponents.

Q: What is the value of $a$?

A: The value of $a$ is $\frac{5}{6}$.

Q: What is the significance of this problem?

A: This problem demonstrates the concept of exponent rules and how to simplify expressions with exponents.

Q: How can this problem be used in real-world applications?

A: This problem can be used to demonstrate the concept of exponent rules and how to simplify expressions with exponents in real-world applications, such as finance, science, and engineering.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include failing to simplify the right-hand side of the equation, not applying the power of a power rule correctly, and not simplifying the exponent correctly.

Q: Where can I find more information on exponent rules and how to simplify expressions with exponents?

A: You can find more information on exponent rules and how to simplify expressions with exponents in the following resources: