What Is The Following Product? Assume $d \geq 0$.$\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}$A. $d$ B. $d^3$ C. $3(\sqrt[3]{d}$\] D. $\sqrt{3d}$

In mathematics, we often come across various types of expressions and equations that require us to simplify or evaluate them. One such expression is given as $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}$, where $d$ is a non-negative number. The question is to determine the value of this expression.

Understanding the Expression

The given expression involves the multiplication of three cube roots of the same number $d$. To simplify this expression, we need to understand the properties of cube roots and their behavior when multiplied together.

Properties of Cube Roots

A cube root of a number $d$ is denoted by $\sqrt[3]{d}$ and is defined as the number that, when multiplied by itself twice, gives the original number $d$. In other words, $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d} = d$.

Simplifying the Expression

Now, let's simplify the given expression by multiplying the three cube roots together:

$\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d} = (\sqrt[3]{d})^3$

Using the property of exponents, we can rewrite the expression as:

$(\sqrt[3]{d})^3 = d^{\frac{3}{3}} = d^1 = d$

Therefore, the value of the given expression is simply $d$.

Conclusion

In conclusion, the value of the expression $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}$ is $d$. This is because the cube root of a number $d$ multiplied by itself three times gives the original number $d$.

Answer

The correct answer is:

A. $d$

Why Not the Other Options?

Now, let's see why the other options are not correct:

• Option B: $d^3$ is incorrect because the cube root of $d$ multiplied by itself three times gives $d$, not $d^3$.
• Option C: $3(\sqrt[3]{d})$ is incorrect because the cube root of $d$ multiplied by itself three times gives $d$, not $3(\sqrt[3]{d})$.
• Option D: $\sqrt{3d}$ is incorrect because the cube root of $d$ multiplied by itself three times gives $d$, not $\sqrt{3d}$.

Final Thoughts

In our previous article, we explored the concept of cube roots and their properties. We simplified the expression $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}$ and found that its value is $d$. In this article, we will answer some frequently asked questions related to cube roots and their properties.

Q: What is a cube root?

A: A cube root of a number $d$ is denoted by $\sqrt[3]{d}$ and is defined as the number that, when multiplied by itself twice, gives the original number $d$. In other words, $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d} = d$.

Q: How do I simplify a cube root expression?

A: To simplify a cube root expression, you can use the property of exponents. For example, if you have the expression $(\sqrt[3]{d})^3$, you can rewrite it as $d^{\frac{3}{3}} = d^1 = d$.

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

A: A square root of a number $d$ is denoted by $\sqrt{d}$ and is defined as the number that, when multiplied by itself, gives the original number $d$. On the other hand, a cube root of a number $d$ is denoted by $\sqrt[3]{d}$ and is defined as the number that, when multiplied by itself twice, gives the original number $d$.

Q: Can I simplify a cube root expression by multiplying it by a number?

A: Yes, you can simplify a cube root expression by multiplying it by a number. For example, if you have the expression $\sqrt[3]{d} \cdot 2$, you can rewrite it as $2\sqrt[3]{d}$.

Q: How do I evaluate a cube root expression with a variable?

A: To evaluate a cube root expression with a variable, you can use the property of exponents. For example, if you have the expression $\sqrt[3]{x^3}$, you can rewrite it as $x$.

Q: Can I use a calculator to evaluate a cube root expression?

A: Yes, you can use a calculator to evaluate a cube root expression. Most calculators have a cube root button that you can use to evaluate cube root expressions.

Q: What are some common mistakes to avoid when working with cube roots?

A: Some common mistakes to avoid when working with cube roots include:

• Not using the property of exponents to simplify cube root expressions
• Not understanding the difference between a cube root and a square root
• Not using a calculator to evaluate cube root expressions
• Not checking your work for errors

Conclusion

In conclusion, cube roots are an important concept in mathematics that can be used to simplify and evaluate expressions. By understanding the properties of cube roots and how to simplify them, you can solve a wide range of mathematical problems. We hope this article has helped you understand cube roots and their properties better.

Common Cube Root Formulas

Here are some common cube root formulas that you can use to simplify and evaluate expressions:

• $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d} = d$
• $(\sqrt[3]{d})^3 = d$
• $\sqrt[3]{d^3} = d$
• $\sqrt[3]{x^3} = x$

Practice Problems

Here are some practice problems that you can use to test your understanding of cube roots and their properties:

• Simplify the expression $\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}$.
• Evaluate the expression $\sqrt[3]{x^3}$.
• Simplify the expression $(\sqrt[3]{d})^3$.
• Evaluate the expression $\sqrt[3]{d^3}$.

We hope these practice problems help you understand cube roots and their properties better.