What Is The Value Of The Expression Below? \left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right ]A. 8 B. 8 \sqrt{8} 8 ​ C. 64 D. 2 8 2 \sqrt{8} 2 8 ​

Understanding Exponents and Powers

In mathematics, exponents and powers are used to represent repeated multiplication of a number. The expression $\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right)$ involves the use of exponents and powers to simplify the expression. To evaluate this expression, we need to understand the properties of exponents and how to apply them.

Properties of Exponents

Exponents have several properties that can be used to simplify expressions. One of the most important properties is the product of powers property, which states that when we multiply two powers with the same base, we can add their exponents. This property can be expressed as:

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

where $a$ is the base and $m$ and $n$ are the exponents.

Applying the Product of Powers Property

In the given expression, we have four terms with the same base, 8, and the same exponent, $1/4$. We can apply the product of powers property to simplify the expression:

$\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) = 8^{1/4 + 1/4 + 1/4 + 1/4}$

Simplifying the Exponent

To simplify the exponent, we can add the fractions by finding a common denominator. In this case, the common denominator is 4. We can rewrite each fraction with a denominator of 4:

$1/4 + 1/4 + 1/4 + 1/4 = 4/4 + 4/4 + 4/4 + 4/4 = 16/4$

Evaluating the Expression

Now that we have simplified the exponent, we can evaluate the expression:

$8^{16/4} = 8^4$

Simplifying the Expression

To simplify the expression, we can evaluate the exponent:

$8^4 = 8 \cdot 8 \cdot 8 \cdot 8 = 4096$

Conclusion

In conclusion, the value of the expression $\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right)$ is 4096.

Answer

The correct answer is A. 4096.

Alternative Solution

Another way to solve this problem is to use the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$. We can rewrite the expression as:

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

Evaluating the Expression

Now that we have rewritten the expression, we can evaluate it:

$\sqrt[4]{8^4} = \sqrt[4]{4096} = 8$

Conclusion

In conclusion, the value of the expression $\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right) \cdot\left(8^{1 / 4}\right)$ is 8.

Answer

The correct answer is A. 8.

Comparison of Solutions

Both solutions arrive at the same answer, 8. However, the first solution uses the product of powers property to simplify the expression, while the second solution uses the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$. Both solutions are valid and can be used to evaluate the expression.

Conclusion

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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 $2^3$, 3 is the exponent and 2 is the base.

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two powers with the same base, we can add their exponents. This property can be expressed as:

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

Q: How do I apply the product of powers property?

A: To apply the product of powers property, we need to identify the base and the exponents in the expression. Then, we can add the exponents and simplify the expression.

Q: What is the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$?

A: This property states that we can rewrite an exponent as a radical by taking the nth root of the base raised to the power of m. For example, $2^{3/4} = \sqrt[4]{2^3}$.

Q: How do I evaluate an expression with a radical?

A: To evaluate an expression with a radical, we need to simplify the radical by finding the nth root of the base raised to the power of m. Then, we can simplify the expression further if possible.

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

A: A positive exponent indicates that we are raising the base to a power, while a negative exponent indicates that we are taking the reciprocal of the base raised to a power. For example, $2^3 = 8$ and $2^{-3} = 1/8$.

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

A: To simplify an expression with a negative exponent, we can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base raised to the power of the absolute value of the exponent. For example, $2^{-3} = 1/2^3$.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $2^0 = 1$.

Q: How do I apply the zero exponent rule?

A: To apply the zero exponent rule, we need to identify the base and the exponent in the expression. If the exponent is zero, we can simplify the expression to 1.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that involves raising a number to a power, while a polynomial expression is an expression that involves adding or subtracting terms with different exponents. For example, $2^3 + 3^2$ is a polynomial expression, while $2^3$ is an exponential expression.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, we need to apply the properties of exponents, such as the product of powers property and the zero exponent rule. We can also rewrite the expression using radicals or other equivalent forms.

Q: What is the importance of understanding exponents and powers?

A: Understanding exponents and powers is important because it allows us to simplify complex expressions and solve problems involving growth and decay. It is also a fundamental concept in algebra and is used extensively in mathematics and science.