Simplify. Assume $z$ Is Greater Than Or Equal To Zero.$2 \sqrt{50 Z^7}$
Understanding the Problem
When dealing with mathematical expressions involving square roots, it's essential to simplify them to make calculations easier. The given expression is $2 \sqrt{50 z^7}$, and we need to simplify it while assuming that $z$ is greater than or equal to zero.
Breaking Down the Expression
To simplify the given expression, we need to break it down into smaller components. We can start by expressing the square root in terms of its prime factors. The expression inside the square root is $50 z^7$, which can be written as $2 \times 5^2 \times z^7$.
Simplifying the Square Root
Now that we have expressed the square root in terms of its prime factors, we can simplify it. The square root of $2 \times 5^2 \times z^7$ can be written as $\sqrt{2} \times \sqrt{5^2} \times \sqrt{z^7}$.
Applying the Properties of Square Roots
We know that the square root of a number raised to a power can be expressed as the number raised to half of that power. Using this property, we can simplify the expression further. The square root of $z^7$ can be written as $z^{7/2}$, which is equivalent to $z^{3.5}$.
Combining the Simplified Components
Now that we have simplified the individual components, we can combine them to get the final simplified expression. The simplified expression is $2 \times \sqrt{2} \times 5 \times z^{3.5}$.
Final Simplified Expression
After combining the simplified components, we get the final simplified expression as $10 \sqrt{2} z^{3.5}$.
Conclusion
In this article, we simplified the given expression $2 \sqrt{50 z^7}$ by breaking it down into smaller components, expressing the square root in terms of its prime factors, and applying the properties of square roots. The final simplified expression is $10 \sqrt{2} z^{3.5}$.
Key Takeaways
- To simplify an expression involving a square root, break it down into smaller components.
- Express the square root in terms of its prime factors.
- Apply the properties of square roots to simplify the expression further.
- Combine the simplified components to get the final simplified expression.
Frequently Asked Questions
- What is the simplified form of $2 \sqrt{50 z^7}$?
- The simplified form is $10 \sqrt{2} z^{3.5}$.
- How do I simplify an expression involving a square root?
- Break it down into smaller components, express the square root in terms of its prime factors, and apply the properties of square roots.
Additional Resources
- For more information on simplifying expressions involving square roots, refer to the following resources:
- Khan Academy: Simplifying Square Roots
- Mathway: Simplifying Square Roots
- Wolfram Alpha: Simplifying Square Roots
Understanding the Problem
When dealing with mathematical expressions involving square roots, it's essential to simplify them to make calculations easier. The given expression is $2 \sqrt{50 z^7}$, and we need to simplify it while assuming that $z$ is greater than or equal to zero.
Breaking Down the Expression
To simplify the given expression, we need to break it down into smaller components. We can start by expressing the square root in terms of its prime factors. The expression inside the square root is $50 z^7$, which can be written as $2 \times 5^2 \times z^7$.
Simplifying the Square Root
Now that we have expressed the square root in terms of its prime factors, we can simplify it. The square root of $2 \times 5^2 \times z^7$ can be written as $\sqrt{2} \times \sqrt{5^2} \times \sqrt{z^7}$.
Applying the Properties of Square Roots
We know that the square root of a number raised to a power can be expressed as the number raised to half of that power. Using this property, we can simplify the expression further. The square root of $z^7$ can be written as $z^{7/2}$, which is equivalent to $z^{3.5}$.
Combining the Simplified Components
Now that we have simplified the individual components, we can combine them to get the final simplified expression. The simplified expression is $2 \times \sqrt{2} \times 5 \times z^{3.5}$.
Final Simplified Expression
After combining the simplified components, we get the final simplified expression as $10 \sqrt{2} z^{3.5}$.
Conclusion
In this article, we simplified the given expression $2 \sqrt{50 z^7}$ by breaking it down into smaller components, expressing the square root in terms of its prime factors, and applying the properties of square roots. The final simplified expression is $10 \sqrt{2} z^{3.5}$.
Key Takeaways
- To simplify an expression involving a square root, break it down into smaller components.
- Express the square root in terms of its prime factors.
- Apply the properties of square roots to simplify the expression further.
- Combine the simplified components to get the final simplified expression.
Frequently Asked Questions
Q: What is the simplified form of $2 \sqrt{50 z^7}$?
A: The simplified form is $10 \sqrt{2} z^{3.5}$.
Q: How do I simplify an expression involving a square root?
A: Break it down into smaller components, express the square root in terms of its prime factors, and apply the properties of square roots.
Q: What is the property of square roots that I can use to simplify the expression?
A: The property of square roots states that the square root of a number raised to a power can be expressed as the number raised to half of that power.
Q: Can I simplify the expression further?
A: Yes, you can simplify the expression further by combining the simplified components.
Q: What is the final simplified expression?
A: The final simplified expression is $10 \sqrt{2} z^{3.5}$.
Additional Resources
- For more information on simplifying expressions involving square roots, refer to the following resources:
- Khan Academy: Simplifying Square Roots
- Mathway: Simplifying Square Roots
- Wolfram Alpha: Simplifying Square Roots
Real-World Applications
Simplifying expressions involving square roots has numerous real-world applications. For example, in physics, you may need to simplify expressions involving square roots to calculate the energy of a particle. In engineering, you may need to simplify expressions involving square roots to calculate the stress on a material.
Conclusion
In conclusion, simplifying expressions involving square roots is an essential skill in mathematics. By breaking down the expression into smaller components, expressing the square root in terms of its prime factors, and applying the properties of square roots, you can simplify the expression and get the final simplified form.
Final Thoughts
Simplifying expressions involving square roots is not just a mathematical concept; it has real-world applications in various fields. By mastering this skill, you can solve complex problems and make calculations easier.