Simplify The Expression:$-4^3 \sqrt{256 X^4} Y^7$

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Understanding the Expression

The given expression is $-4^3 \sqrt{256 x^4} y^7$. To simplify this expression, we need to understand the properties of exponents and radicals. The expression involves a cube of a number, a square root, and variables raised to certain powers.

Breaking Down the Expression

Let's break down the expression into its components:

• $-4^3$ is a negative number raised to the power of 3.
• $\sqrt{256 x^4}$ is the square root of a product of a number and a variable raised to the power of 4.
• $y^7$ is a variable raised to the power of 7.

Simplifying the Cube of a Number

The first step is to simplify the cube of the number $-4^3$. Using the property of exponents, we can rewrite $-4^3$ as $-(4^3)$. This simplifies to $-64$.

Simplifying the Square Root

Next, we need to simplify the square root of the product $256 x^4$. The square root of 256 is 16, and the square root of $x^4$ is $x^2$. Therefore, the square root of $256 x^4$ is $16 x^2$.

Combining the Simplified Components

Now that we have simplified the cube of the number and the square root, we can combine the simplified components. The expression becomes $-64 \sqrt{256 x^4} y^7$. Substituting the simplified square root, we get $-64 (16 x^2) y^7$.

Further Simplification

We can further simplify the expression by multiplying the numbers and combining the variables. $-64 (16 x^2) y^7$ simplifies to $-1024 x^2 y^7$.

Final Simplified Expression

The final simplified expression is $-1024 x^2 y^7$. This is the simplest form of the given expression.

Conclusion

In this article, we simplified the expression $-4^3 \sqrt{256 x^4} y^7$ by breaking it down into its components, simplifying each component, and combining the simplified components. The final simplified expression is $-1024 x^2 y^7$.

Frequently Asked Questions

• What is the simplified form of $-4^3 \sqrt{256 x^4} y^7$?
• How do you simplify the cube of a number?
• How do you simplify the square root of a product?
• What is the final simplified expression?

Answers

• The simplified form of $-4^3 \sqrt{256 x^4} y^7$ is $-1024 x^2 y^7$.
• To simplify the cube of a number, you can use the property of exponents to rewrite the number as a product of a number and itself raised to the power of 3.
• To simplify the square root of a product, you can use the property of radicals to rewrite the square root as the product of the square roots of the individual components.
• The final simplified expression is $-1024 x^2 y^7$.

Step-by-Step Solution

1. Simplify the cube of the number $-4^3$.
2. Simplify the square root of the product $256 x^4$.
3. Combine the simplified components.
4. Further simplify the expression by multiplying the numbers and combining the variables.
5. The final simplified expression is $-1024 x^2 y^7$.

Tips and Tricks

• When simplifying expressions, it's essential to break them down into their components and simplify each component separately.
• Use the properties of exponents and radicals to simplify expressions.
• Combine the simplified components to get the final simplified expression.

Real-World Applications

• Simplifying expressions is a crucial skill in mathematics and has numerous real-world applications, such as in physics, engineering, and computer science.
• Understanding the properties of exponents and radicals is essential in simplifying expressions and solving mathematical problems.

Conclusion

In conclusion, simplifying the expression $-4^3 \sqrt{256 x^4} y^7$ involves breaking it down into its components, simplifying each component, and combining the simplified components. The final simplified expression is $-1024 x^2 y^7$. This article provides a step-by-step solution, tips and tricks, and real-world applications of simplifying expressions.

Frequently Asked Questions

Q1: What is the simplified form of $-4^3 \sqrt{256 x^4} y^7$?

A1: The simplified form of $-4^3 \sqrt{256 x^4} y^7$ is $-1024 x^2 y^7$.

Q2: How do you simplify the cube of a number?

A2: To simplify the cube of a number, you can use the property of exponents to rewrite the number as a product of a number and itself raised to the power of 3.

Q3: How do you simplify the square root of a product?

A3: To simplify the square root of a product, you can use the property of radicals to rewrite the square root as the product of the square roots of the individual components.

Q4: What is the final simplified expression?

A4: The final simplified expression is $-1024 x^2 y^7$.

Q5: How do you combine the simplified components?

A5: To combine the simplified components, you can multiply the numbers and combine the variables.

Q6: What are the properties of exponents and radicals?

A6: The properties of exponents and radicals are essential in simplifying expressions and solving mathematical problems. Exponents are used to represent repeated multiplication, while radicals are used to represent the square root of a number.

Q7: How do you use the properties of exponents and radicals to simplify expressions?

A7: To use the properties of exponents and radicals to simplify expressions, you can break down the expression into its components, simplify each component separately, and combine the simplified components.

Q8: What are the real-world applications of simplifying expressions?

A8: The real-world applications of simplifying expressions include physics, engineering, and computer science.

Q9: How do you apply the properties of exponents and radicals in real-world problems?

A9: To apply the properties of exponents and radicals in real-world problems, you can use the properties to simplify expressions and solve mathematical problems.

Q10: What are some tips and tricks for simplifying expressions?

A10: Some tips and tricks for simplifying expressions include breaking down the expression into its components, using the properties of exponents and radicals, and combining the simplified components.

Additional Questions and Answers

Q11: How do you simplify the expression $-3^2 \sqrt{81 x^6} y^8$?

A11: To simplify the expression $-3^2 \sqrt{81 x^6} y^8$, you can break it down into its components, simplify each component separately, and combine the simplified components. The simplified expression is $-27 x^3 y^8$.

Q12: How do you simplify the expression $-2^4 \sqrt{16 x^8} y^9$?

A12: To simplify the expression $-2^4 \sqrt{16 x^8} y^9$, you can break it down into its components, simplify each component separately, and combine the simplified components. The simplified expression is $-64 x^4 y^9$.

Q13: How do you simplify the expression $-5^3 \sqrt{25 x^{10}} y^{11}$?

A13: To simplify the expression $-5^3 \sqrt{25 x^{10}} y^{11}$, you can break it down into its components, simplify each component separately, and combine the simplified components. The simplified expression is $-125 x^5 y^{11}$.

Conclusion

In conclusion, simplifying expressions is a crucial skill in mathematics and has numerous real-world applications. By understanding the properties of exponents and radicals, you can simplify expressions and solve mathematical problems. This article provides a Q&A section to help you understand the concepts and apply them in real-world problems.