Evaluate The Expression:$\[ \sqrt[3]{-64} + \sqrt{64} \\]

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Introduction

In mathematics, expressions involving radicals and exponents are common and can be challenging to evaluate. The given expression, −643+64\sqrt[3]{-64} + \sqrt{64}, requires us to find the cube root of -64 and the square root of 64, and then add the results together. In this article, we will break down the steps to evaluate this expression and provide a clear understanding of the mathematical concepts involved.

Understanding Radicals and Exponents

Before we dive into the evaluation of the expression, let's briefly review the concepts of radicals and exponents. A radical is a mathematical expression that involves a root, such as a square root or cube root. The square root of a number, denoted by x\sqrt{x}, is a value that, when multiplied by itself, gives the original number. Similarly, the cube root of a number, denoted by x3\sqrt[3]{x}, is a value that, when cubed, gives the original number.

Exponents, on the other hand, are mathematical operations that involve raising a number to a power. For example, 232^3 means 2 raised to the power of 3, which is equal to 8.

Evaluating the Cube Root of -64

To evaluate the cube root of -64, we need to find a number that, when cubed, gives -64. We can start by listing the perfect cubes of negative numbers:

  • (−1)3=−1(-1)^3 = -1
  • (−2)3=−8(-2)^3 = -8
  • (−3)3=−27(-3)^3 = -27
  • (−4)3=−64(-4)^3 = -64

As we can see, the cube root of -64 is -4, since (−4)3=−64(-4)^3 = -64.

Evaluating the Square Root of 64

To evaluate the square root of 64, we need to find a number that, when squared, gives 64. We can start by listing the perfect squares of positive numbers:

  • 12=11^2 = 1
  • 22=42^2 = 4
  • 32=93^2 = 9
  • 42=164^2 = 16
  • 52=255^2 = 25
  • 62=366^2 = 36
  • 72=497^2 = 49
  • 82=648^2 = 64

As we can see, the square root of 64 is 8, since 82=648^2 = 64.

Adding the Results Together

Now that we have evaluated the cube root of -64 and the square root of 64, we can add the results together:

−643+64=−4+8=4\sqrt[3]{-64} + \sqrt{64} = -4 + 8 = 4

Therefore, the final answer to the expression −643+64\sqrt[3]{-64} + \sqrt{64} is 4.

Conclusion

In this article, we evaluated the expression −643+64\sqrt[3]{-64} + \sqrt{64} by finding the cube root of -64 and the square root of 64, and then adding the results together. We reviewed the concepts of radicals and exponents, and provided a clear understanding of the mathematical operations involved. By following the steps outlined in this article, readers should be able to evaluate similar expressions involving radicals and exponents.

Frequently Asked Questions

  • Q: What is the cube root of -64? A: The cube root of -64 is -4, since (−4)3=−64(-4)^3 = -64.
  • Q: What is the square root of 64? A: The square root of 64 is 8, since 82=648^2 = 64.
  • Q: What is the final answer to the expression −643+64\sqrt[3]{-64} + \sqrt{64}? A: The final answer to the expression −643+64\sqrt[3]{-64} + \sqrt{64} is 4.

Further Reading

For more information on radicals and exponents, readers may want to consult the following resources:

  • Khan Academy: Radicals and Exponents
  • Mathway: Radicals and Exponents
  • Wolfram MathWorld: Radicals and Exponents

By following the steps outlined in this article and consulting the recommended resources, readers should be able to evaluate expressions involving radicals and exponents with confidence.

Introduction

In our previous article, we evaluated the expression −643+64\sqrt[3]{-64} + \sqrt{64} by finding the cube root of -64 and the square root of 64, and then adding the results together. In this article, we will answer some frequently asked questions (FAQs) related to evaluating expressions involving radicals and exponents.

Q&A

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

A: A radical is a mathematical expression that involves a root, such as a square root or cube root. An exponent, on the other hand, is a mathematical operation that involves raising a number to a power.

Q: How do I evaluate an expression involving a cube root?

A: To evaluate an expression involving a cube root, you need to find a number that, when cubed, gives the original number. For example, to evaluate −643\sqrt[3]{-64}, you need to find a number that, when cubed, gives -64.

Q: How do I evaluate an expression involving a square root?

A: To evaluate an expression involving a square root, you need to find a number that, when squared, gives the original number. For example, to evaluate 64\sqrt{64}, you need to find a number that, when squared, gives 64.

Q: What is the order of operations for evaluating expressions involving radicals and exponents?

A: The order of operations for evaluating expressions involving radicals and exponents is:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponents (such as squaring or cubing).
  3. Evaluate any radicals (such as square roots or cube roots).
  4. Perform any multiplication and division operations from left to right.
  5. Perform any addition and subtraction operations from left to right.

Q: How do I simplify an expression involving a radical?

A: To simplify an expression involving a radical, you need to find the largest perfect square or perfect cube that divides the radicand (the number inside the radical). For example, to simplify 16\sqrt{16}, you can rewrite it as 4×4\sqrt{4 \times 4}, which is equal to 414\sqrt{1}, or simply 4.

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

A: A rational number is a number that can be expressed as the ratio of two integers, such as 3/4 or 22/7. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers, such as the square root of 2 or the cube root of 3.

Q: How do I determine whether a number is rational or irrational?

A: To determine whether a number is rational or irrational, you need to check whether it can be expressed as the ratio of two integers. If it can, then it is rational. If it cannot, then it is irrational.

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions involving radicals and exponents. We covered topics such as the difference between radicals and exponents, how to evaluate expressions involving cube roots and square roots, and how to simplify expressions involving radicals. By following the steps outlined in this article, readers should be able to evaluate expressions involving radicals and exponents with confidence.

Frequently Asked Questions

  • Q: What is the difference between a radical and an exponent? A: A radical is a mathematical expression that involves a root, such as a square root or cube root. An exponent, on the other hand, is a mathematical operation that involves raising a number to a power.
  • Q: How do I evaluate an expression involving a cube root? A: To evaluate an expression involving a cube root, you need to find a number that, when cubed, gives the original number.
  • Q: How do I evaluate an expression involving a square root? A: To evaluate an expression involving a square root, you need to find a number that, when squared, gives the original number.

Further Reading

For more information on radicals and exponents, readers may want to consult the following resources:

  • Khan Academy: Radicals and Exponents
  • Mathway: Radicals and Exponents
  • Wolfram MathWorld: Radicals and Exponents

By following the steps outlined in this article and consulting the recommended resources, readers should be able to evaluate expressions involving radicals and exponents with confidence.