What Is The Value Of The Expression Below?$\left(8^{2 / 3}\right)^{1 / 2}$A. 8 B. 4 C. 2 D. 1

Understanding Exponents and Powers

In mathematics, exponents and powers are used to represent repeated multiplication of a number. The expression $\left(8^{2 / 3}\right)^{1 / 2}$ 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 Rules of Exponents

When dealing with exponents, there are several rules to keep in mind:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent Rule: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Evaluating the Expression

Now that we have a good understanding of the rules of exponents, let's apply them to the given expression $\left(8^{2 / 3}\right)^{1 / 2}$.

First, we need to simplify the inner exponent $2/3$. To do this, we can rewrite $8$ as $2^3$, since $8 = 2^3$. Then, we can apply the power of a power rule to get:

$$\left(8^{2 / 3}\right)^{1 / 2} = \left((2^3)^{2 / 3}\right)^{1 / 2}$$

Next, we can simplify the inner exponent $2/3$ by applying the power of a power rule:

$$\left((2^3)^{2 / 3}\right)^{1 / 2} = (2^3)^{(2/3) \cdot (1/2)}$$

Now, we can simplify the exponent $(2/3) \cdot (1/2)$ by multiplying the fractions:

$$(2/3) \cdot (1/2) = 2/6 = 1/3$$

So, the expression becomes:

$$(2^3)^{(2/3) \cdot (1/2)} = (2^3)^{1/3}$$

Finally, we can apply the power of a power rule to get:

$$(2^3)^{1/3} = 2^{3 \cdot (1/3)} = 2^1 = 2$$

Conclusion

In conclusion, the value of the expression $\left(8^{2 / 3}\right)^{1 / 2}$ is $2$. This is because we simplified the expression by applying the rules of exponents, including the product of powers rule, power of a power rule, zero exponent rule, and negative exponent rule.

Answer

The correct answer is:

• C. 2

Additional Examples

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

• Example 1: Evaluate the expression $(2^4)^{1/2}$.
• Example 2: Evaluate the expression $(3^2)^{-1}$.
• Example 3: Evaluate the expression $(4^3)^{2/3}$.

By applying the rules of exponents and powers, we can simplify these expressions and find their values.

Final Thoughts

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Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about exponents and powers. Whether you're a student, teacher, or simply someone looking to brush up on their math skills, this Q&A section is for you.

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

A: An exponent is a small number that is written to the right of a base number, indicating how many times the base number should be multiplied by itself. A power, on the other hand, is the result of raising a base number to a certain exponent.

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

A: To simplify an expression with multiple exponents, you can use the product of powers rule, which states that when multiplying two powers with the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the rule for negative exponents?

A: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

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

A: To evaluate an expression with a fractional exponent, you can rewrite the base number as a power of a smaller base number. For example, $a^{m/n} = (a^m)^{1/n}$.

Q: What is the rule for zero exponents?

A: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q: How do I simplify an expression with a power of a power?

A: To simplify an expression with a power of a power, you can use the power of a power rule, which states that when raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

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

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant. A power function, on the other hand, is a function of the form $f(x) = x^n$, where $n$ is a constant.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or graph paper. The graph of an exponential function will be a curve that increases or decreases rapidly as the input value increases or decreases.

Q: What are some common applications of exponents and powers?

A: Exponents and powers have many real-world applications, including finance, science, and engineering. For example, compound interest is calculated using exponents, and the growth of populations can be modeled using power functions.

Q: How do I use exponents and powers in real-world problems?

A: To use exponents and powers in real-world problems, you can apply the rules of exponents and powers to simplify complex expressions and find their values. For example, you can use exponents to calculate the growth of populations or the value of investments.

Conclusion

In this Q&A article, we've answered some of the most frequently asked questions about exponents and powers. Whether you're a student, teacher, or simply someone looking to brush up on their math skills, this article is for you. By understanding the rules of exponents and powers, you can simplify complex expressions and find their values, and apply these concepts to real-world problems.