What Is The Following Product? Assume Y ≥ 0 Y \geq 0 Y ≥ 0 . Y 3 ⋅ Y 3 \sqrt{y^3} \cdot \sqrt{y^3} Y 3 ​ ⋅ Y 3 ​ A. Y 3 Y^3 Y 3 B. 2 Y 3 2y^3 2 Y 3 C. Y 6 Y^6 Y 6 D. 2 Y 6 2y^6 2 Y 6

Understanding the Problem

The given problem involves simplifying an expression that contains square roots and exponents. We are asked to find the product of two square roots, each containing the variable $y$ raised to the power of 3. The expression is $\sqrt{y^3} \cdot \sqrt{y^3}$, and we need to simplify it to determine the correct answer.

Simplifying the Expression

To simplify the expression, we can start by using the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property to the given expression, we get:

$$\sqrt{y^3} \cdot \sqrt{y^3} = \sqrt{y^3 \cdot y^3}$$

Using Exponent Rules

Next, we can use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. Applying this rule to the expression, we get:

$$\sqrt{y^3 \cdot y^3} = \sqrt{y^{3+3}}$$

Simplifying the Exponent

Now, we can simplify the exponent by adding the two exponents together:

$$\sqrt{y^{3+3}} = \sqrt{y^6}$$

Final Answer

The final answer is $\sqrt{y^6}$. However, we need to express this in a simplified form. We can do this by using the property of square roots that states $\sqrt{a^2} = a$. Applying this property to the expression, we get:

$$\sqrt{y^6} = y^3$$

But Wait!

We are not done yet. We need to consider the original expression and the options provided. The original expression is $\sqrt{y^3} \cdot \sqrt{y^3}$, and we simplified it to $y^3$. However, we need to consider the possibility that the expression can be further simplified.

Revisiting the Expression

Let's revisit the expression and see if we can simplify it further. We can start by using the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property to the expression, we get:

$$\sqrt{y^3} \cdot \sqrt{y^3} = \sqrt{y^3 \cdot y^3}$$

Using Exponent Rules Again

Next, we can use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. Applying this rule to the expression, we get:

$$\sqrt{y^3 \cdot y^3} = \sqrt{y^{3+3}}$$

Simplifying the Exponent Again

Now, we can simplify the exponent by adding the two exponents together:

$$\sqrt{y^{3+3}} = \sqrt{y^6}$$

Final Answer (Again!)

The final answer is $\sqrt{y^6}$. However, we need to express this in a simplified form. We can do this by using the property of square roots that states $\sqrt{a^2} = a$. Applying this property to the expression, we get:

$$\sqrt{y^6} = y^3$$

But What About the Options?

We are given four options: A. $y^3$, B. $2y^3$, C. $y^6$, and D. $2y^6$. We simplified the expression to $y^3$, but we need to consider the possibility that the expression can be further simplified.

Comparing the Options

Let's compare the options and see which one matches our simplified expression. We have:

• A. $y^3$
• B. $2y^3$
• C. $y^6$
• D. $2y^6$

The Correct Answer

The correct answer is A. $y^3$. This is because our simplified expression is $y^3$, which matches option A.

Conclusion

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Frequently Asked Questions

We have received many questions about the problem of simplifying the expression $\sqrt{y^3} \cdot \sqrt{y^3}$. Here are some of the most frequently asked questions and their answers:

Q: What is the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$?

A: This property states that the product of two square roots is equal to the square root of the product of the two numbers inside the square roots.

Q: How do I simplify the expression $\sqrt{y^3} \cdot \sqrt{y^3}$?

A: To simplify the expression, you can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Then, you can use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the exponent.

Q: What is the final answer to the problem?

A: The final answer is $\sqrt{y^6}$. However, this can be simplified further to $y^3$.

Q: Why is the correct answer A. $y^3$?

A: The correct answer is A. $y^3$ because our simplified expression is $y^3$, which matches option A.

Q: Can the expression $\sqrt{y^3} \cdot \sqrt{y^3}$ be further simplified?

A: Yes, the expression can be further simplified. We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ and the rule of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression.

Q: What is the difference between the options A. $y^3$ and B. $2y^3$?

A: The difference between the options A. $y^3$ and B. $2y^3$ is that option A. $y^3$ is the simplified expression, while option B. $2y^3$ is not the simplified expression.

Q: Why is option C. $y^6$ not the correct answer?

A: Option C. $y^6$ is not the correct answer because our simplified expression is $y^3$, not $y^6$.

Q: Why is option D. $2y^6$ not the correct answer?

A: Option D. $2y^6$ is not the correct answer because our simplified expression is $y^3$, not $2y^6$.

Common Mistakes

Here are some common mistakes that students make when simplifying the expression $\sqrt{y^3} \cdot \sqrt{y^3}$:

• Not using the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
• Not using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$.
• Simplifying the expression to $y^6$ instead of $y^3$.
• Not comparing the options to determine the correct answer.

Tips and Tricks

Here are some tips and tricks that can help you simplify the expression $\sqrt{y^3} \cdot \sqrt{y^3}$:

• Use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify the expression.
• Use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the exponent.
• Compare the options to determine the correct answer.
• Make sure to simplify the expression to the simplest form possible.

Conclusion

In conclusion, the correct answer to the problem is A. $y^3$. We simplified the expression using the properties of square roots and exponents, and we compared the options to determine the correct answer. We also discussed common mistakes and tips and tricks that can help you simplify the expression.