Which Choice Is Equivalent To The Product Below? 2 ⋅ 3 ⋅ 8 \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8} 2 ​ ⋅ 3 ​ ⋅ 8 ​ A. 8 12 8 \sqrt{12} 8 12 ​ B. 4 12 4 \sqrt{12} 4 12 ​ C. 4 3 4 \sqrt{3} 4 3 ​ D. 16 3 16 \sqrt{3} 16 3 ​

Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of radicals and how to simplify them. In this article, we'll explore the concept of simplifying radical expressions and apply it to the given problem: $\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8}$.

The Product of Radicals

The product of radicals is a fundamental concept in mathematics that allows us to simplify complex expressions. When multiplying radicals, we can combine the radicals by multiplying the numbers inside the radicals. In this case, we have:

$\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8}$

To simplify this expression, we can start by multiplying the numbers inside the radicals:

$\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8} = \sqrt{2 \cdot 3 \cdot 8}$

Simplifying the Expression

Now that we have the product of the numbers inside the radicals, we can simplify the expression further. We can start by multiplying the numbers:

$2 \cdot 3 \cdot 8 = 48$

So, the expression becomes:

$\sqrt{48}$

Simplifying the Square Root

The square root of 48 can be simplified by finding the largest perfect square that divides 48. In this case, the largest perfect square that divides 48 is 16. We can write 48 as:

$48 = 16 \cdot 3$

Now, we can simplify the square root:

$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$

Comparing the Simplified Expression to the Options

Now that we have simplified the expression, we can compare it to the options:

A. $8 \sqrt{12}$ B. $4 \sqrt{12}$ C. $4 \sqrt{3}$ D. $16 \sqrt{3}$

The simplified expression is $4 \sqrt{3}$, which matches option C.

Conclusion

In this article, we explored the concept of simplifying radical expressions and applied it to the given problem. We started by multiplying the radicals and then simplified the expression by finding the largest perfect square that divides the product. Finally, we compared the simplified expression to the options and found that the correct answer is option C: $4 \sqrt{3}$.

Key Takeaways

• The product of radicals can be simplified by multiplying the numbers inside the radicals.
• The square root of a product can be simplified by finding the largest perfect square that divides the product.
• Simplifying radical expressions is an essential skill in mathematics that can be applied to a wide range of problems.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Final Answer

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Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root. For example, $\sqrt{2}$ is a radical expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can start by multiplying the numbers inside the radical. Then, you can simplify the expression by finding the largest perfect square that divides the product.

Q: What is the difference between a square root and a radical?

A: A square root is a specific type of radical that represents the number that, when multiplied by itself, gives a specified value. For example, $\sqrt{4}$ is a square root because $2 \cdot 2 = 4$. A radical is a more general term that includes square roots and other roots.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. For example, $\sqrt{2x}$ can be simplified by finding the largest perfect square that divides $2x$. In this case, the largest perfect square that divides $2x$ is $2x$, so the expression can be simplified to $\sqrt{2x}$.

Q: How do I simplify a radical expression with a coefficient?

A: To simplify a radical expression with a coefficient, you can start by multiplying the coefficient by the number inside the radical. Then, you can simplify the expression by finding the largest perfect square that divides the product.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. For example, $\sqrt{-4}$ can be simplified by finding the largest perfect square that divides $-4$. In this case, the largest perfect square that divides $-4$ is $-4$, so the expression can be simplified to $2i$, where $i$ is the imaginary unit.

Q: How do I simplify a radical expression with a fraction?

A: To simplify a radical expression with a fraction, you can start by multiplying the numerator and denominator by the radical. Then, you can simplify the expression by finding the largest perfect square that divides the product.

Q: Can I simplify a radical expression with a decimal?

A: Yes, you can simplify a radical expression with a decimal. For example, $\sqrt{2.5}$ can be simplified by finding the largest perfect square that divides $2.5$. In this case, the largest perfect square that divides $2.5$ is $2.5$, so the expression can be simplified to $1.581138830\ldots$.

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

A: To simplify a radical expression with a negative exponent, you can start by rewriting the expression with a positive exponent. Then, you can simplify the expression by finding the largest perfect square that divides the product.

Conclusion

In this article, we've answered some of the most frequently asked questions about simplifying radical expressions. We've covered topics such as simplifying radical expressions with variables, coefficients, negative numbers, fractions, decimals, and negative exponents. By following these tips and techniques, you'll be able to simplify radical expressions with ease.

Key Takeaways

• Simplifying radical expressions is an essential skill in mathematics that can be applied to a wide range of problems.
• To simplify a radical expression, you can start by multiplying the numbers inside the radical.
• You can simplify a radical expression with a variable, coefficient, negative number, fraction, decimal, or negative exponent.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Final Answer

The final answer is: Simplifying radical expressions is a fundamental concept in mathematics that can be applied to a wide range of problems.