What Is The Value Of The Expression Below?$\left(4^2\right)^{1 / 4}$A. 2 B. 4 C. 1 D. 8

Understanding Exponents and Powers

In mathematics, exponents and powers are used to represent repeated multiplication of a number. The expression $\left(4^2\right)^{1 / 4}$ involves both exponents and powers, making it a bit more complex to evaluate. In this article, we will break down the expression and determine its value.

Breaking Down the Expression

The given expression is $\left(4^2\right)^{1 / 4}$. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent inside the parentheses: $4^2 = 16$
2. Raise the result to the power of $1/4$: $16^{1/4}$

Evaluating the Power

Now that we have the result of the exponent inside the parentheses, we can evaluate the power. To raise a number to a fractional power, we can use the following property:

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

In this case, we have $16^{1/4} = \sqrt[4]{16^1}$

Simplifying the Expression

To simplify the expression, we can evaluate the fourth root of $16^1$. Since $16 = 2^4$, we can rewrite the expression as:

$\sqrt[4]{16^1} = \sqrt[4]{(2^4)^1} = \sqrt[4]{2^4}$

Evaluating the Fourth Root

Now that we have the expression in a simpler form, we can evaluate the fourth root. The fourth root of a number is the number that, when raised to the power of 4, gives the original number. In this case, we have:

$\sqrt[4]{2^4} = 2$

Conclusion

In conclusion, the value of the expression $\left(4^2\right)^{1 / 4}$ is 2. This is because we evaluated the exponent inside the parentheses, raised the result to the power of $1/4$, and simplified the expression using the properties of exponents and powers.

Common Mistakes to Avoid

When evaluating expressions with exponents and powers, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations (PEMDAS)
• Not simplifying the expression using the properties of exponents and powers
• Not evaluating the exponent inside the parentheses correctly

Practice Problems

To practice evaluating expressions with exponents and powers, try the following problems:

1. Evaluate the expression $\left(3^2\right)^{1 / 3}$
2. Evaluate the expression $\left(2^3\right)^{1 / 2}$
3. Evaluate the expression $\left(5^4\right)^{1 / 5}$

Answer Key

1. $\sqrt[3]{9}$
2. $\sqrt{8}$
3. $\sqrt[5]{625}$

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about evaluating expressions with exponents and powers.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: To evaluate an expression with a fractional exponent, you can use the following property:

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

For example, to evaluate $16^{1/4}$, you can rewrite it as $\sqrt[4]{16^1}$.

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

A: A power and an exponent are related but distinct concepts. An exponent is a number that is raised to a power. For example, in the expression $2^3$, the 3 is an exponent and the 2 is the base. A power, on the other hand, is the result of raising a number to a power. For example, in the expression $2^3$, the result is 8, which is a power of 2.

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

A: To simplify an expression with multiple exponents, you can use the following property:

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

For example, to simplify the expression $(2^3)^2$, you can rewrite it as $2^{3 \cdot 2} = 2^6$.

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

A: A positive exponent and a negative exponent are related but distinct concepts. A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as the reciprocal. For example, in the expression $2^3$, the 3 is a positive exponent and the result is 8. In the expression $2^{-3}$, the -3 is a negative exponent and the result is 1/8.

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

A: To evaluate an expression with a zero exponent, you can use the following property:

$a^0 = 1$

For example, to evaluate $2^0$, you can rewrite it as 1.

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

A: A rational exponent and an irrational exponent are related but distinct concepts. A rational exponent is a fraction or a decimal, while an irrational exponent is a number that cannot be expressed as a fraction or a decimal. For example, in the expression $2^{1/2}$, the 1/2 is a rational exponent and the result is the square root of 2. In the expression $2^{\sqrt{2}}$, the $\sqrt{2}$ is an irrational exponent and the result is a number that cannot be expressed as a fraction or a decimal.

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

A: To evaluate an expression with a mixed exponent, you can use the following property:

$a^{m/n} \cdot a^p = a^{m/n + p}$

For example, to evaluate the expression $2^{1/2} \cdot 2^3$, you can rewrite it as $2^{1/2 + 3} = 2^{7/2}$.

Conclusion

In conclusion, evaluating expressions with exponents and powers can be challenging, but with practice and patience, you can become proficient in evaluating these types of expressions. Remember to follow the order of operations (PEMDAS), simplify the expression using the properties of exponents and powers, and evaluate the exponent inside the parentheses correctly. With these skills, you'll be able to tackle even the most complex expressions with confidence.