Evaluate The Following. Click On Not A Real Number If Applicable.(a) \[$-\sqrt[4]{81} =\$\] \[$\square\$\](b) \[$\sqrt[3]{-64} =\$\]

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Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will evaluate two radical expressions and determine whether they are real numbers or not.

Evaluating Radical Expressions

A radical expression is a mathematical expression that contains a root or a power of a number. The most common types of radical expressions are square roots, cube roots, and fourth roots. To evaluate a radical expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Evaluating Expression (a)

The first radical expression we need to evaluate is:

−814-\sqrt[4]{81}

To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the fourth root: 814\sqrt[4]{81}
  2. Simplify the expression: 814=344=3\sqrt[4]{81} = \sqrt[4]{3^4} = 3
  3. Multiply the result by -1: −3-3

Therefore, the value of the expression −814-\sqrt[4]{81} is -3.

Evaluating Expression (b)

The second radical expression we need to evaluate is:

−643\sqrt[3]{-64}

To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the cube root: −643\sqrt[3]{-64}
  2. Simplify the expression: −643=−433=−4\sqrt[3]{-64} = \sqrt[3]{-4^3} = -4
  3. The expression is a real number.

Therefore, the value of the expression −643\sqrt[3]{-64} is -4.

Conclusion

In conclusion, we have evaluated two radical expressions and determined their values. The first expression, −814-\sqrt[4]{81}, has a value of -3, while the second expression, −643\sqrt[3]{-64}, has a value of -4. Both expressions are real numbers.

Final Answer

Based on our evaluation, we can conclude that:

  • Expression (a) −814-\sqrt[4]{81} is a real number.
  • Expression (b) −643\sqrt[3]{-64} is a real number.

Therefore, the final answer is:

  • (a) -3
  • (b) -4
    Evaluating Radical Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we evaluated two radical expressions and determined their values. However, we understand that there may be more questions and concerns regarding radical expressions. In this article, we will address some of the most frequently asked questions about radical expressions.

Q&A

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number. The most common types of radical expressions are square roots, cube roots, and fourth roots.

Q: How do I evaluate a radical expression?

A: To evaluate a radical expression, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: A square root is a root of a number that is raised to the power of 1/2, while a cube root is a root of a number that is raised to the power of 1/3.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to look for any perfect squares or perfect cubes that can be factored out of the expression.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be expressed as 2^3.

Q: How do I handle negative numbers in radical expressions?

A: When a negative number is inside a radical expression, you need to consider the sign of the result. If the index of the radical is even, the result will be positive. If the index of the radical is odd, the result will be negative.

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

A: Yes, you can simplify a radical expression with a negative number inside. However, you need to consider the sign of the result.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, while an irrational number is a number that cannot be expressed as the ratio of two integers.

Q: Can a radical expression be a rational or irrational number?

A: Yes, a radical expression can be a rational or irrational number, depending on the value of the expression.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions about radical expressions. We hope that this article has provided you with a better understanding of radical expressions and how to evaluate them.

Final Tips

  • Always follow the order of operations when evaluating a radical expression.
  • Look for any perfect squares or perfect cubes that can be factored out of the expression.
  • Consider the sign of the result when a negative number is inside a radical expression.
  • A radical expression can be a rational or irrational number, depending on the value of the expression.

Common Mistakes

  • Not following the order of operations when evaluating a radical expression.
  • Not looking for any perfect squares or perfect cubes that can be factored out of the expression.
  • Not considering the sign of the result when a negative number is inside a radical expression.

Final Answer

Based on our Q&A guide, we can conclude that:

  • A radical expression is a mathematical expression that contains a root or a power of a number.
  • To evaluate a radical expression, you need to follow the order of operations (PEMDAS).
  • A perfect square is a number that can be expressed as the square of an integer.
  • A perfect cube is a number that can be expressed as the cube of an integer.
  • A radical expression can be a rational or irrational number, depending on the value of the expression.