Evaluate The Expression: 16 3 + 54 3 \sqrt[3]{16} + \sqrt[3]{54} 3 16 ​ + 3 54 ​

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Introduction

In mathematics, evaluating expressions involving radicals is a crucial skill that requires a deep understanding of the properties of exponents and radicals. The given expression, $\sqrt[3]{16} + \sqrt[3]{54}$, involves the sum of two cube roots. To evaluate this expression, we need to simplify each cube root individually and then add them together. In this article, we will break down the process of evaluating the given expression and provide a step-by-step solution.

Simplifying the Cube Roots

To simplify the cube roots, we need to find the cube root of each number individually. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In other words, $\sqrt[3]{x} = y$ means that $y^3 = x$.

Simplifying $\sqrt[3]{16}$

To simplify $\sqrt[3]{16}$, we need to find a number that, when multiplied by itself twice, gives 16. We can start by listing the perfect cubes of numbers:

• $1^3 = 1$
• $2^3 = 8$
• $3^3 = 27$

Since 16 is between 8 and 27, we can conclude that $\sqrt[3]{16}$ is between 2 and 3. However, we need to find the exact value. We can rewrite 16 as a product of 8 and 2:

$$16 = 8 \cdot 2$$

Now, we can take the cube root of both sides:

$$\sqrt[3]{16} = \sqrt[3]{8 \cdot 2}$$

Using the property of radicals that $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, we can rewrite the expression as:

$$\sqrt[3]{16} = \sqrt[3]{8} \cdot \sqrt[3]{2}$$

Since $\sqrt[3]{8} = 2$, we can simplify the expression further:

$$\sqrt[3]{16} = 2 \cdot \sqrt[3]{2}$$

Simplifying $\sqrt[3]{54}$

To simplify $\sqrt[3]{54}$, we need to find a number that, when multiplied by itself twice, gives 54. We can start by listing the perfect cubes of numbers:

• $1^3 = 1$
• $2^3 = 8$
• $3^3 = 27$
• $4^3 = 64$

Since 54 is between 27 and 64, we can conclude that $\sqrt[3]{54}$ is between 3 and 4. However, we need to find the exact value. We can rewrite 54 as a product of 27 and 2:

$$54 = 27 \cdot 2$$

Now, we can take the cube root of both sides:

$$\sqrt[3]{54} = \sqrt[3]{27 \cdot 2}$$

Using the property of radicals that $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, we can rewrite the expression as:

$$\sqrt[3]{54} = \sqrt[3]{27} \cdot \sqrt[3]{2}$$

Since $\sqrt[3]{27} = 3$, we can simplify the expression further:

$$\sqrt[3]{54} = 3 \cdot \sqrt[3]{2}$$

Adding the Simplified Cube Roots

Now that we have simplified both cube roots, we can add them together:

$$\sqrt[3]{16} + \sqrt[3]{54} = 2 \cdot \sqrt[3]{2} + 3 \cdot \sqrt[3]{2}$$

Using the distributive property of multiplication over addition, we can rewrite the expression as:

$$\sqrt[3]{16} + \sqrt[3]{54} = (2 + 3) \cdot \sqrt[3]{2}$$

Simplifying the expression further, we get:

$$\sqrt[3]{16} + \sqrt[3]{54} = 5 \cdot \sqrt[3]{2}$$

Conclusion

In this article, we evaluated the expression $\sqrt[3]{16} + \sqrt[3]{54}$ by simplifying each cube root individually and then adding them together. We used the properties of radicals and exponents to simplify the expressions and arrived at the final answer of $5 \cdot \sqrt[3]{2}$. This problem requires a deep understanding of the properties of radicals and exponents, and it is an excellent example of how to evaluate expressions involving cube roots.

Frequently Asked Questions

• Q: What is the value of $\sqrt[3]{16}$? A: The value of $\sqrt[3]{16}$ is $2 \cdot \sqrt[3]{2}$.
• Q: What is the value of $\sqrt[3]{54}$? A: The value of $\sqrt[3]{54}$ is $3 \cdot \sqrt[3]{2}$.
• Q: What is the value of $\sqrt[3]{16} + \sqrt[3]{54}$? A: The value of $\sqrt[3]{16} + \sqrt[3]{54}$ is $5 \cdot \sqrt[3]{2}$.

Further Reading

• Radical Expressions: A radical expression is an expression that contains a radical, which is a symbol that represents the square root or cube root of a number.
• Properties of Radicals: The properties of radicals include the product rule, quotient rule, and power rule.
• Exponents: Exponents are a shorthand way of writing repeated multiplication. For example, $2^3$ means $2 \cdot 2 \cdot 2$.

References

• [1] "Radical Expressions and Equations" by Paul A. Foerster
• [2] "Algebra and Trigonometry" by Michael Sullivan
• [3] "Mathematics for the Nonmathematician" by Morris Kline
  

Introduction

In our previous article, we evaluated the expression $\sqrt[3]{16} + \sqrt[3]{54}$ by simplifying each cube root individually and then adding them together. In this article, we will answer some frequently asked questions related to evaluating expressions involving cube roots.

Q&A

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

A: A cube root is a value that, when multiplied by itself twice, gives the original number. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$. A square root is a value that, when multiplied by itself, gives the original number. For example, $\sqrt{4} = 2$ because $2^2 = 4$.

Q: How do I simplify a cube root?

A: To simplify a cube root, you need to find a number that, when multiplied by itself twice, gives the original number. You can start by listing the perfect cubes of numbers and then use the properties of radicals to simplify the expression.

Q: What is the product rule for radicals?

A: The product rule for radicals states that $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. This means that you can separate the product of two numbers into the product of their cube roots.

Q: What is the quotient rule for radicals?

A: The quotient rule for radicals states that $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$. This means that you can separate the quotient of two numbers into the quotient of their cube roots.

Q: How do I add or subtract cube roots?

A: To add or subtract cube roots, you need to combine the numbers inside the cube roots. For example, $\sqrt[3]{16} + \sqrt[3]{54} = 2 \cdot \sqrt[3]{2} + 3 \cdot \sqrt[3]{2} = (2 + 3) \cdot \sqrt[3]{2} = 5 \cdot \sqrt[3]{2}$.

Q: What is the power rule for radicals?

A: The power rule for radicals states that $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$. This means that you can raise a cube root to a power by raising the number inside the cube root to that power.

Q: How do I evaluate an expression involving cube roots and exponents?

A: To evaluate an expression involving cube roots and exponents, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside the cube roots, then evaluate any exponents, and finally add or subtract the results.

Examples

Example 1: Evaluating $\sqrt[3]{27} + \sqrt[3]{64}$

To evaluate this expression, we need to simplify each cube root individually and then add them together.

$$\sqrt[3]{27} = 3$$

$$\sqrt[3]{64} = 4$$

$$\sqrt[3]{27} + \sqrt[3]{64} = 3 + 4 = 7$$

Example 2: Evaluating $\sqrt[3]{8} \cdot \sqrt[3]{27}$

To evaluate this expression, we need to use the product rule for radicals.

$$\sqrt[3]{8} \cdot \sqrt[3]{27} = \sqrt[3]{8 \cdot 27}$$

$$\sqrt[3]{8 \cdot 27} = \sqrt[3]{216}$$

$$\sqrt[3]{216} = 6$$

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions involving cube roots. We covered topics such as simplifying cube roots, using the product rule and quotient rule for radicals, and evaluating expressions involving cube roots and exponents. We also provided examples to illustrate the concepts.

Frequently Asked Questions

• Q: What is the value of $\sqrt[3]{16}$? A: The value of $\sqrt[3]{16}$ is $2 \cdot \sqrt[3]{2}$.
• Q: What is the value of $\sqrt[3]{54}$? A: The value of $\sqrt[3]{54}$ is $3 \cdot \sqrt[3]{2}$.
• Q: What is the value of $\sqrt[3]{16} + \sqrt[3]{54}$? A: The value of $\sqrt[3]{16} + \sqrt[3]{54}$ is $5 \cdot \sqrt[3]{2}$.

Further Reading

• Radical Expressions: A radical expression is an expression that contains a radical, which is a symbol that represents the square root or cube root of a number.
• Properties of Radicals: The properties of radicals include the product rule, quotient rule, and power rule.
• Exponents: Exponents are a shorthand way of writing repeated multiplication. For example, $2^3$ means $2 \cdot 2 \cdot 2$.

References

• [1] "Radical Expressions and Equations" by Paul A. Foerster
• [2] "Algebra and Trigonometry" by Michael Sullivan
• [3] "Mathematics for the Nonmathematician" by Morris Kline