Simplify: − 64 3 + 5 4 2 + 3 2 \sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}} 4 2 + 3 2 3 − 64 ​ + 5 ​ ​

Introduction

Mathematics is a vast and complex subject that involves the study of numbers, quantities, and shapes. It is a fundamental tool for problem-solving and critical thinking, and it has numerous applications in various fields, including science, engineering, economics, and finance. In this article, we will focus on simplifying a mathematical expression involving radicals and fractions.

Understanding the Expression

The given expression is $\sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}}$. To simplify this expression, we need to understand the individual components and how they interact with each other. The expression involves a cube root, a square root, and a fraction.

Breaking Down the Components

• The cube root of -64 is $\sqrt[3]{-64} = -4$.
• The square of 4 is $4^2 = 16$.
• The square of 3 is $3^2 = 9$.
• The sum of 16 and 9 is $16 + 9 = 25$.

Simplifying the Expression

Now that we have broken down the components, we can simplify the expression.

Simplifying the Fraction

The fraction is $\frac{\sqrt[3]{-64}+5}{4^2+3^2}$. We can substitute the values we found earlier to get:

$\frac{-4+5}{25} = \frac{1}{25}$

Simplifying the Square Root

The square root of $\frac{1}{25}$ is $\sqrt{\frac{1}{25}} = \frac{1}{5}$

Conclusion

In conclusion, the simplified expression is $\frac{1}{5}$. This is the final answer to the given problem.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Break down the components of the expression:
   • The cube root of -64 is $\sqrt[3]{-64} = -4$.
   • The square of 4 is $4^2 = 16$.
   • The square of 3 is $3^2 = 9$.
   • The sum of 16 and 9 is $16 + 9 = 25$.
2. Simplify the fraction:
   • The fraction is $\frac{\sqrt[3]{-64}+5}{4^2+3^2}$.
   • We can substitute the values we found earlier to get: $\frac{-4+5}{25} = \frac{1}{25}$.
3. Simplify the square root:
   • The square root of $\frac{1}{25}$ is $\sqrt{\frac{1}{25}} = \frac{1}{5}$.

Frequently Asked Questions

Here are some frequently asked questions related to the problem:

• Q: What is the cube root of -64? A: The cube root of -64 is -4.
• Q: What is the square of 4? A: The square of 4 is 16.
• Q: What is the square of 3? A: The square of 3 is 9.
• Q: What is the sum of 16 and 9? A: The sum of 16 and 9 is 25.

Final Answer

The final answer to the problem is $\frac{1}{5}$.

Introduction

In our previous article, we simplified the mathematical expression $\sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}}$. In this article, we will provide a Q&A section to address some common questions and doubts related to the problem.

Q&A

Q: What is the cube root of -64?

A: The cube root of -64 is -4. This is because -4 is the number that, when multiplied by itself three times, gives -64.

Q: How do you simplify the cube root of -64?

A: To simplify the cube root of -64, you can use the fact that the cube root of a negative number is the negative of the cube root of its absolute value. In this case, the cube root of -64 is -4, which is the negative of the cube root of 64.

Q: What is the square of 4?

A: The square of 4 is 16. This is because 4 multiplied by itself gives 16.

Q: What is the square of 3?

A: The square of 3 is 9. This is because 3 multiplied by itself gives 9.

Q: How do you simplify the fraction $\frac{\sqrt[3]{-64}+5}{4^2+3^2}$?

A: To simplify the fraction, you can substitute the values of the cube root of -64 and the squares of 4 and 3. In this case, the fraction becomes $\frac{-4+5}{16+9} = \frac{1}{25}$.

Q: How do you simplify the square root of $\frac{1}{25}$?

A: To simplify the square root of $\frac{1}{25}$, you can use the fact that the square root of a fraction is the square root of the numerator divided by the square root of the denominator. In this case, the square root of $\frac{1}{25}$ is $\frac{1}{5}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\frac{1}{5}$.

Common Mistakes

Here are some common mistakes to avoid when simplifying the expression:

• Not simplifying the cube root of -64 correctly.
• Not substituting the values of the squares of 4 and 3 correctly.
• Not simplifying the fraction correctly.
• Not simplifying the square root of $\frac{1}{25}$ correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify the expression:

• Make sure to simplify the cube root of -64 correctly.
• Use the fact that the cube root of a negative number is the negative of the cube root of its absolute value.
• Substitute the values of the squares of 4 and 3 correctly.
• Simplify the fraction correctly by dividing the numerator by the denominator.
• Simplify the square root of $\frac{1}{25}$ correctly by dividing the numerator by the square root of the denominator.

Conclusion

In conclusion, simplifying the expression $\sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}}$ requires careful attention to detail and a thorough understanding of the concepts involved. By following the steps outlined in this article and avoiding common mistakes, you can simplify the expression correctly and arrive at the final answer of $\frac{1}{5}$.