What Is The Value Of The Expression Below? ( 4 1 / 2 ) 2 \left(4^{1 / 2}\right)^2 ( 4 1/2 ) 2 A. 16 B. 8 C. 4 D. 12

Understanding Exponents and Powers

In mathematics, exponents and powers are used to represent repeated multiplication of a number. The expression $\left(4^{1 / 2}\right)^2$ involves both exponents and powers, making it a bit complex to evaluate. To simplify this expression, we need to understand the rules of exponents and how to handle fractional exponents.

The Rule of Exponents

The rule of exponents states that when we have a power raised to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \cdot n}$. This rule is essential in simplifying expressions involving exponents.

Simplifying the Expression

Let's start by simplifying the expression $\left(4^{1 / 2}\right)^2$. According to the rule of exponents, we can rewrite this expression as $4^{(1/2) \cdot 2}$.

Evaluating the Exponent

Now, let's evaluate the exponent $(1/2) \cdot 2$. Multiplying a fraction by a whole number is equivalent to multiplying the numerator by that number. Therefore, $(1/2) \cdot 2 = 1 \cdot 2 = 2$.

Simplifying the Expression Further

Now that we have evaluated the exponent, we can simplify the expression $4^{(1/2) \cdot 2}$ as $4^2$.

Evaluating the Expression

Finally, let's evaluate the expression $4^2$. According to the definition of exponentiation, $4^2$ means multiplying 4 by itself 2 times. Therefore, $4^2 = 4 \cdot 4 = 16$.

Conclusion

In conclusion, the value of the expression $\left(4^{1 / 2}\right)^2$ is 16. This result is obtained by applying the rule of exponents and simplifying the expression step by step.

Answer

The correct answer is A. 16.

Additional Examples

To further illustrate the concept of exponents and powers, let's consider a few more examples.

Example 1

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

Solution

Using the rule of exponents, we can rewrite this expression as $3^{(1/3) \cdot 3}$. Evaluating the exponent, we get $(1/3) \cdot 3 = 1 \cdot 3 = 3$. Therefore, the expression simplifies to $3^3$, which equals $3 \cdot 3 \cdot 3 = 27$.

Example 2

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

Solution

Using the rule of exponents, we can rewrite this expression as $2^{(1/4) \cdot 4}$. Evaluating the exponent, we get $(1/4) \cdot 4 = 1 \cdot 4 = 4$. Therefore, the expression simplifies to $2^4$, which equals $2 \cdot 2 \cdot 2 \cdot 2 = 16$.

Example 3

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

Solution

Using the rule of exponents, we can rewrite this expression as $5^{(1/5) \cdot 5}$. Evaluating the exponent, we get $(1/5) \cdot 5 = 1 \cdot 5 = 5$. Therefore, the expression simplifies to $5^5$, which equals $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 3125$.

Final Thoughts

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Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about exponents and powers.

Q1: What is the rule of exponents?

A1: The rule of exponents states that when we have a power raised to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \cdot n}$.

Q2: How do I simplify an expression with exponents?

A2: To simplify an expression with exponents, we need to apply the rule of exponents. We can rewrite the expression as a power raised to another power, and then multiply the exponents.

Q3: What is the difference between a power and an exponent?

A3: A power is the base number that is being raised to a certain exponent. An exponent is the number that is being multiplied by the base number.

Q4: How do I evaluate an expression with a fractional exponent?

A4: To evaluate an expression with a fractional exponent, we need to multiply the base number by itself as many times as the denominator of the fraction. For example, $4^{1/2} = 4 \cdot 4 = 16$.

Q5: Can I simplify an expression with a negative exponent?

A5: Yes, we can simplify an expression with a negative exponent by rewriting it as a positive exponent. For example, $4^{-2} = 1/4^2 = 1/16$.

Q6: How do I evaluate an expression with a zero exponent?

A6: Any number raised to the power of zero is equal to 1. For example, $4^0 = 1$.

Q7: Can I simplify an expression with a variable exponent?

A7: Yes, we can simplify an expression with a variable exponent by applying the rule of exponents. For example, $(x^m)^n = x^{m \cdot n}$.

Q8: How do I evaluate an expression with a mixed exponent?

A8: To evaluate an expression with a mixed exponent, we need to apply the rule of exponents and simplify the expression step by step. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q9: Can I simplify an expression with a fractional base?

A9: Yes, we can simplify an expression with a fractional base by rewriting it as a power of a whole number. For example, $(1/2)^3 = (1/2) \cdot (1/2) \cdot (1/2) = 1/8$.

Q10: How do I evaluate an expression with a negative base?

A10: To evaluate an expression with a negative base, we need to apply the rule of exponents and simplify the expression step by step. For example, $(-2)^3 = -2 \cdot -2 \cdot -2 = -8$.

Conclusion

In conclusion, exponents and powers are fundamental concepts in mathematics that can be used to simplify complex expressions. By applying the rule of exponents and understanding the properties of exponents, we can evaluate expressions with ease. We hope that this Q&A article has provided you with a better understanding of exponents and powers.

Additional Resources

For more information on exponents and powers, we recommend the following resources:

• Khan Academy: Exponents and Powers
• Mathway: Exponents and Powers
• Wolfram Alpha: Exponents and Powers

Final Thoughts

In conclusion, exponents and powers are essential concepts in mathematics that can be used to simplify complex expressions. By applying the rule of exponents and understanding the properties of exponents, we can evaluate expressions with ease. We hope that this Q&A article has provided you with a better understanding of exponents and powers.