What Is The Value Of The Expression Below?$\left(49^3\right)^{1 / 6}$A. 7 B. 49 C. 24.5 D. 3.5

Understanding the Expression

The given expression is $\left(49^3\right)^{1 / 6}$. To evaluate this expression, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we have an exponentiation within an exponentiation.

Simplifying the Expression

To simplify the expression, we can start by evaluating the inner exponentiation. We have $49^3$, which means $49$ raised to the power of $3$. This can be calculated as follows:

$49^3 = 49 \times 49 \times 49 = 117649$

Now that we have the value of $49^3$, we can substitute it back into the original expression:

$\left(49^3\right)^{1 / 6} = \left(117649\right)^{1 / 6}$

Evaluating the Outer Exponentiation

Now, we need to evaluate the outer exponentiation, which is $\left(117649\right)^{1 / 6}$. To do this, we can raise $117649$ to the power of $\frac{1}{6}$.

$\left(117649\right)^{1 / 6} = \sqrt[6]{117649}$

Calculating the Sixth Root

To calculate the sixth root of $117649$, we can use a calculator or estimate it manually. The sixth root of $117649$ is approximately $7$.

$\sqrt[6]{117649} \approx 7$

Conclusion

Therefore, the value of the expression $\left(49^3\right)^{1 / 6}$ is approximately $7$.

Alternative Solution

We can also simplify the expression using the property of exponents that states $(a^m)^n = a^{mn}$. Applying this property to the given expression, we get:

$\left(49^3\right)^{1 / 6} = 49^{3 \times \frac{1}{6}} = 49^{\frac{1}{2}} = \sqrt{49} = 7$

This alternative solution also yields the value of $7$ for the expression.

Final Answer

The final answer is $\boxed{7}$.

Discussion

The given expression can be evaluated using the order of operations and the properties of exponents. The expression can be simplified by evaluating the inner exponentiation and then the outer exponentiation. The alternative solution uses the property of exponents to simplify the expression. Both solutions yield the value of $7$ for the expression.

Related Topics

• Order of operations
• Properties of exponents
• Simplifying expressions
• Evaluating expressions

Practice Problems

• Evaluate the expression $\left(8^2\right)^{1 / 3}$
• Simplify the expression $\left(27^4\right)^{1 / 2}$
• Evaluate the expression $\left(16^3\right)^{1 / 4}$
  

Frequently Asked Questions

Q: What is the order of operations when evaluating expressions with exponents?

A: The order of operations when evaluating expressions with exponents is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the property of exponents that states $(a^m)^n = a^{mn}$. This property allows you to combine the exponents and simplify the expression.

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

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3$, the $3$ is the exponent and the $2$ is the base.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$. This property allows you to rewrite the expression with a positive exponent.

Q: What is the value of the expression $\left(64^2\right)^{1 / 4}$?

A: To evaluate this expression, we can start by evaluating the inner exponentiation. We have $64^2$, which means $64$ raised to the power of $2$. This can be calculated as follows:

$64^2 = 64 \times 64 = 4096$

Now that we have the value of $64^2$, we can substitute it back into the original expression:

$\left(64^2\right)^{1 / 4} = \left(4096\right)^{1 / 4}$

Next, we can evaluate the outer exponentiation, which is $\left(4096\right)^{1 / 4}$. To do this, we can raise $4096$ to the power of $\frac{1}{4}$.

$\left(4096\right)^{1 / 4} = \sqrt[4]{4096} = 8$

Therefore, the value of the expression $\left(64^2\right)^{1 / 4}$ is $8$.

Q: What is the value of the expression $\left(27^3\right)^{1 / 3}$?

A: To evaluate this expression, we can start by evaluating the inner exponentiation. We have $27^3$, which means $27$ raised to the power of $3$. This can be calculated as follows:

$27^3 = 27 \times 27 \times 27 = 19683$

Now that we have the value of $27^3$, we can substitute it back into the original expression:

$\left(27^3\right)^{1 / 3} = \left(19683\right)^{1 / 3}$

Next, we can evaluate the outer exponentiation, which is $\left(19683\right)^{1 / 3}$. To do this, we can raise $19683$ to the power of $\frac{1}{3}$.

$\left(19683\right)^{1 / 3} = \sqrt[3]{19683} = 27$

Therefore, the value of the expression $\left(27^3\right)^{1 / 3}$ is $27$.

Q: What is the value of the expression $\left(16^4\right)^{1 / 4}$?

A: To evaluate this expression, we can start by evaluating the inner exponentiation. We have $16^4$, which means $16$ raised to the power of $4$. This can be calculated as follows:

$16^4 = 16 \times 16 \times 16 \times 16 = 65536$

Now that we have the value of $16^4$, we can substitute it back into the original expression:

$\left(16^4\right)^{1 / 4} = \left(65536\right)^{1 / 4}$

Next, we can evaluate the outer exponentiation, which is $\left(65536\right)^{1 / 4}$. To do this, we can raise $65536$ to the power of $\frac{1}{4}$.

$\left(65536\right)^{1 / 4} = \sqrt[4]{65536} = 16$

Therefore, the value of the expression $\left(16^4\right)^{1 / 4}$ is $16$.

Final Answer

The final answer is $\boxed{16}$.

Discussion

The given expressions can be evaluated using the order of operations and the properties of exponents. The expressions can be simplified by evaluating the inner exponentiation and then the outer exponentiation. The alternative solutions use the properties of exponents to simplify the expressions. Both solutions yield the value of $16$ for the expression.

Related Topics

• Order of operations
• Properties of exponents
• Simplifying expressions
• Evaluating expressions

Practice Problems

• Evaluate the expression $\left(8^2\right)^{1 / 3}$
• Simplify the expression $\left(27^4\right)^{1 / 2}$
• Evaluate the expression $\left(16^3\right)^{1 / 4}$