Simplify. Assume $b$ Is Greater Than 0. 75 B 10 \sqrt{75 B^{10}} 75 B 10 ​

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Understanding the Problem

When dealing with square roots, it's essential to simplify the expression to make it easier to work with. In this case, we're given the expression $\sqrt{75 b^{10}}$, and we need to simplify it. To start, let's break down the expression and identify the components that can be simplified.

Breaking Down the Expression

The expression $\sqrt{75 b^{10}}$ can be broken down into two parts: the coefficient $75$ and the variable part $b^{10}$. The coefficient $75$ can be further broken down into its prime factors, which are $3 \times 5^2$. The variable part $b^{10}$ can be simplified by using the property of exponents that states $\sqrt{x^n} = x^{n/2}$.

Simplifying the Coefficient

Let's start by simplifying the coefficient $75$. We can rewrite it as $3 \times 5^2$. Since the square root of $5^2$ is $5$, we can simplify the coefficient as follows:

$$\sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3}$$

Simplifying the Variable Part

Now that we've simplified the coefficient, let's move on to the variable part $b^{10}$. Using the property of exponents that states $\sqrt{x^n} = x^{n/2}$, we can simplify the variable part as follows:

$$\sqrt{b^{10}} = b^{10/2} = b^5$$

Combining the Simplified Parts

Now that we've simplified the coefficient and the variable part, we can combine them to get the final simplified expression:

$$\sqrt{75 b^{10}} = 5\sqrt{3} \times b^5 = 5b^5\sqrt{3}$$

Conclusion

In this article, we simplified the expression $\sqrt{75 b^{10}}$ by breaking it down into its components and using the properties of exponents and square roots. We simplified the coefficient $75$ by rewriting it as $3 \times 5^2$ and taking the square root of the prime factors. We also simplified the variable part $b^{10}$ by using the property of exponents that states $\sqrt{x^n} = x^{n/2}$. By combining the simplified parts, we arrived at the final simplified expression $5b^5\sqrt{3}$.

Additional Tips and Tricks

When dealing with square roots, it's essential to remember the following tips and tricks:

• Simplify the coefficient: Break down the coefficient into its prime factors and take the square root of the prime factors.
• Use the property of exponents: Use the property of exponents that states $\sqrt{x^n} = x^{n/2}$ to simplify the variable part.
• Combine the simplified parts: Combine the simplified coefficient and variable part to get the final simplified expression.

Real-World Applications

Simplifying expressions like $\sqrt{75 b^{10}}$ has real-world applications in various fields, including:

• Mathematics: Simplifying expressions is a fundamental concept in mathematics, and it's used extensively in algebra, geometry, and calculus.
• Physics: Simplifying expressions is used in physics to solve problems involving motion, energy, and forces.
• Engineering: Simplifying expressions is used in engineering to design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

Simplifying expressions like $\sqrt{75 b^{10}}$ may seem like a trivial task, but it's an essential skill that's used extensively in various fields. By breaking down the expression into its components and using the properties of exponents and square roots, we can simplify the expression and arrive at the final simplified expression $5b^5\sqrt{3}$.

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Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying the expression $\sqrt{75 b^{10}}$.

Q: What is the first step in simplifying the expression $\sqrt{75 b^{10}}$?

A: The first step in simplifying the expression $\sqrt{75 b^{10}}$ is to break down the coefficient $75$ into its prime factors.

Q: How do I break down the coefficient $75$ into its prime factors?

A: To break down the coefficient $75$ into its prime factors, we can use the following steps:

1. Divide $75$ by the smallest prime number, which is $2$.
2. Since $75$ is not divisible by $2$, we move on to the next prime number, which is $3$.
3. We can divide $75$ by $3$ to get $25$.
4. Since $25$ is not a prime number, we can further break it down into its prime factors, which are $5^2$.

Q: What is the simplified form of the coefficient $75$?

A: The simplified form of the coefficient $75$ is $3 \times 5^2$.

Q: How do I simplify the variable part $b^{10}$?

A: To simplify the variable part $b^{10}$, we can use the property of exponents that states $\sqrt{x^n} = x^{n/2}$.

Q: What is the simplified form of the variable part $b^{10}$?

A: The simplified form of the variable part $b^{10}$ is $b^5$.

Q: How do I combine the simplified coefficient and variable part to get the final simplified expression?

A: To combine the simplified coefficient and variable part, we can multiply them together.

Q: What is the final simplified expression?

A: The final simplified expression is $5b^5\sqrt{3}$.

Q: What are some real-world applications of simplifying expressions like $\sqrt{75 b^{10}}$?

A: Simplifying expressions like $\sqrt{75 b^{10}}$ has real-world applications in various fields, including mathematics, physics, and engineering.

Q: Why is it important to simplify expressions like $\sqrt{75 b^{10}}$?

A: Simplifying expressions like $\sqrt{75 b^{10}}$ is important because it makes it easier to work with the expression and can help us arrive at the final answer more quickly.

Q: Can I use a calculator to simplify expressions like $\sqrt{75 b^{10}}$?

A: While a calculator can be useful for simplifying expressions, it's not always necessary. In many cases, we can simplify expressions like $\sqrt{75 b^{10}}$ using basic algebraic manipulations.

Additional Resources

If you're struggling to simplify expressions like $\sqrt{75 b^{10}}$, here are some additional resources that may be helpful:

• Algebra textbooks: There are many algebra textbooks that provide step-by-step instructions for simplifying expressions like $\sqrt{75 b^{10}}$.
• Online resources: There are many online resources, such as Khan Academy and Mathway, that provide video lessons and interactive exercises for simplifying expressions like $\sqrt{75 b^{10}}$.
• Math tutors: If you're really struggling to simplify expressions like $\sqrt{75 b^{10}}$, consider hiring a math tutor who can provide one-on-one instruction and support.

Conclusion

Simplifying expressions like $\sqrt{75 b^{10}}$ may seem like a challenging task, but with practice and patience, you can master the skills you need to simplify even the most complex expressions. Remember to break down the coefficient into its prime factors, use the property of exponents to simplify the variable part, and combine the simplified parts to get the final simplified expression. With these skills, you'll be able to tackle even the most challenging math problems with confidence.