Which Choices Are Equivalent To The Expression Below? Check All That Apply. $4 \sqrt{3}$A. $\sqrt{24} \cdot \sqrt{2}$B. $3 \sqrt{16}$C. $\sqrt{4} \cdot \sqrt{3}$D. $\sqrt{12} = \sqrt{4}$E. 48F.

Which Choices are Equivalent to the Expression Below?

The given expression is $4 \sqrt{3}$. To find equivalent choices, we need to understand the properties of radicals and how they can be manipulated. A radical is a symbol used to represent the square root of a number. In this case, $\sqrt{3}$ represents the square root of 3.

Choice A: $\sqrt{24} \cdot \sqrt{2}$

To determine if this choice is equivalent to the given expression, we need to simplify the radicals. We can start by factoring the numbers inside the radicals.

$\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6}$

Now, we can multiply this result by $\sqrt{2}$.

$2 \sqrt{6} \cdot \sqrt{2} = 2 \sqrt{12}$

Since $\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}$, we can simplify this expression further.

$2 \sqrt{12} = 2 \cdot 2 \sqrt{3} = 4 \sqrt{3}$

This choice is equivalent to the given expression.

Choice B: $3 \sqrt{16}$

To determine if this choice is equivalent to the given expression, we need to simplify the radicals. We can start by factoring the numbers inside the radicals.

$\sqrt{16} = \sqrt{4^2} = 4$

Now, we can multiply this result by 3.

$3 \cdot 4 = 12$

This choice is not equivalent to the given expression.

Choice C: $\sqrt{4} \cdot \sqrt{3}$

To determine if this choice is equivalent to the given expression, we need to simplify the radicals. We can start by factoring the numbers inside the radicals.

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

Now, we can multiply this result by $\sqrt{3}$.

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

This choice is not equivalent to the given expression.

Choice D: $\sqrt{12} = \sqrt{4}$

To determine if this choice is equivalent to the given expression, we need to simplify the radicals. We can start by factoring the numbers inside the radicals.

$\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}$

This choice is not equivalent to the given expression.

Choice E: 48

To determine if this choice is equivalent to the given expression, we need to simplify the expression. We can start by multiplying the numbers together.

$4 \cdot 12 = 48$

This choice is not equivalent to the given expression.

Based on the analysis above, only Choice A: $\sqrt{24} \cdot \sqrt{2}$ is equivalent to the given expression $4 \sqrt{3}$. The other choices do not simplify to the same expression.

• When simplifying radicals, we need to factor the numbers inside the radicals.
• We can use the properties of radicals to simplify expressions.
• Not all choices will be equivalent to the given expression.

Try simplifying the following expressions:

• $\sqrt{36} \cdot \sqrt{5}$
• $\sqrt{9} \cdot \sqrt{16}$
• $\sqrt{25} \cdot \sqrt{4}$

Use the properties of radicals to simplify these expressions and determine if they are equivalent to the given expression.
Q&A: Radicals and Simplification

Radicals and simplification are fundamental concepts in mathematics, particularly in algebra and geometry. In this article, we will answer some common questions related to radicals and simplification.

Q: What is a radical?

A radical is a symbol used to represent the square root of a number. It is denoted by the symbol $\sqrt{}$. For example, $\sqrt{4}$ represents the square root of 4.

Q: How do I simplify a radical?

To simplify a radical, we need to factor the number inside the radical. We can use the properties of radicals to simplify expressions. For example, $\sqrt{36} = \sqrt{6^2} = 6$.

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

A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. A radical is used to represent the square root of a number, while an exponent is used to represent repeated multiplication. For example, $2^3 = 8$ and $\sqrt{8} = 2\sqrt{2}$.

Q: Can I simplify a radical with a variable?

Yes, you can simplify a radical with a variable. For example, $\sqrt{16x} = \sqrt{4^2x} = 4\sqrt{x}$.

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

To simplify a radical with a coefficient, we need to factor the coefficient and the number inside the radical. For example, $3\sqrt{12} = 3\sqrt{4 \cdot 3} = 3 \cdot 2 \sqrt{3} = 6\sqrt{3}$.

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

No, you cannot simplify a radical with a negative number. Radicals are only defined for non-negative numbers.

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

To simplify a radical with a fraction, we need to factor the numerator and the denominator. For example, $\sqrt{\frac{12}{16}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2}$.

Q: Can I simplify a radical with a decimal?

No, you cannot simplify a radical with a decimal. Radicals are only defined for integers and fractions.

Radicals and simplification are fundamental concepts in mathematics. By understanding how to simplify radicals, you can solve a wide range of problems in algebra and geometry. Remember to factor the number inside the radical, use the properties of radicals, and simplify expressions to get the final answer.

Try simplifying the following expressions:

• $\sqrt{20}$
• $\sqrt{9x}$
• $\sqrt{\frac{16}{25}}$
• $\sqrt{2.5}$

Use the properties of radicals to simplify these expressions and determine if they are equivalent to the given expression.