Simplify This Expression Completely: 4 A 10 25 B 6 \sqrt{\frac{4 A^{10}}{25 B^6}} 25 B 6 4 A 10 ​ ​ INSTRUCTIONS:- Use The / Symbol For Division If Necessary. For Example, For 1 X \frac{1}{x} X 1 ​ , Type 1 / X 1 / X 1/ X .- Use Brackets To Put Multiple Factors In The

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression $\sqrt{\frac{4 a^{10}}{25 b^6}}$ using the rules of exponents and radicals.

Understanding the Expression

The given expression is $\sqrt{\frac{4 a^{10}}{25 b^6}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have a fraction under the square root sign, which means we need to simplify the fraction first.

Simplifying the Fraction

To simplify the fraction, we can start by factoring the numerator and denominator. The numerator is $4 a^{10}$, and the denominator is $25 b^6$. We can factor out the greatest common factor (GCF) of the numerator and denominator, which is $a^2 b^3$.

\frac{4 a^{10}}{25 b^6} = \frac{a^2 b^3 \cdot 2 a^8}{5^2 b^6}

Applying the Quotient Rule

Now that we have factored the numerator and denominator, we can apply the quotient rule of exponents. The quotient rule states that when we divide two powers with the same base, we subtract the exponents.

\frac{a^2 b^3 \cdot 2 a^8}{5^2 b^6} = \frac{2 a^{10} b^{-3}}{5^2}

Simplifying the Radical

Now that we have simplified the fraction, we can simplify the radical. The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

\sqrt{\frac{2 a^{10} b^{-3}}{5^2}} = \frac{\sqrt{2 a^{10} b^{-3}}}{\sqrt{5^2}}

Applying the Product Rule

To simplify the numerator, we can apply the product rule of radicals. The product rule states that when we multiply two radicals, we multiply the numbers inside the radicals.

\sqrt{2 a^{10} b^{-3}} = \sqrt{2} \cdot \sqrt{a^{10}} \cdot \sqrt{b^{-3}}

Applying the Power Rule

To simplify the expression further, we can apply the power rule of radicals. The power rule states that when we raise a radical to a power, we raise the number inside the radical to that power.

\sqrt{a^{10}} = a^5

Simplifying the Expression

Now that we have simplified the numerator, we can simplify the expression.

\frac{\sqrt{2} \cdot a^5 \cdot b^{-3/2}}{5} = \frac{\sqrt{2} a^5 b^{-3/2}}{5}

Conclusion

In this article, we simplified the expression $\sqrt{\frac{4 a^{10}}{25 b^6}}$ using the rules of exponents and radicals. We factored the numerator and denominator, applied the quotient rule, simplified the radical, applied the product rule, and applied the power rule to simplify the expression. The final simplified expression is $\frac{\sqrt{2} a^5 b^{-3/2}}{5}$.

Final Answer

The final answer is $\boxed{\frac{\sqrt{2} a^5 b^{-3/2}}{5}}$.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

FAQs

Q: What is the square root of a fraction? A: The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

Q: How do I simplify a radical expression? A: To simplify a radical expression, you can factor the numerator and denominator, apply the quotient rule, simplify the radical, apply the product rule, and apply the power rule.

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will provide a Q&A guide to help you understand and simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root of a number. For example, $\sqrt{4}$ is a radical expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can follow these steps:

1. Factor the numerator and denominator.
2. Apply the quotient rule.
3. Simplify the radical.
4. Apply the product rule.
5. Apply the power rule.

Q: What is the quotient rule?

A: The quotient rule states that when we divide two powers with the same base, we subtract the exponents. For example, $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$.

Q: What is the product rule?

A: The product rule states that when we multiply two radicals, we multiply the numbers inside the radicals. For example, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: What is the power rule?

A: The power rule states that when we raise a radical to a power, we raise the number inside the radical to that power. For example, $(\sqrt{a})^2 = a$.

Q: How do I simplify a fraction under a square root?

A: To simplify a fraction under a square root, you can follow these steps:

1. Factor the numerator and denominator.
2. Apply the quotient rule.
3. Simplify the radical.
4. Apply the product rule.
5. Apply the power rule.

Q: What is the final simplified expression for $\sqrt{\frac{4 a^{10}}{25 b^6}}$?

A: The final simplified expression is $\frac{\sqrt{2} a^5 b^{-3/2}}{5}$.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent. To do this, you can apply the power rule and simplify the expression.

Q: How do I simplify a radical expression with a variable in the denominator?

A: To simplify a radical expression with a variable in the denominator, you can follow these steps:

1. Factor the numerator and denominator.
2. Apply the quotient rule.
3. Simplify the radical.
4. Apply the product rule.
5. Apply the power rule.

Q: Can I simplify a radical expression with a fraction in the numerator?

A: Yes, you can simplify a radical expression with a fraction in the numerator. To do this, you can follow the same steps as simplifying a fraction under a square root.

Q: How do I simplify a radical expression with multiple terms in the numerator?

A: To simplify a radical expression with multiple terms in the numerator, you can follow these steps:

1. Factor the numerator and denominator.
2. Apply the quotient rule.
3. Simplify the radical.
4. Apply the product rule.
5. Apply the power rule.

Conclusion

In this article, we provided a Q&A guide to help you understand and simplify radical expressions. We covered topics such as the quotient rule, product rule, power rule, and simplifying fractions under a square root. We also provided examples and final simplified expressions to help you practice and understand the concepts.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

FAQs

Q: What is the square root of a fraction? A: The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

Q: How do I simplify a radical expression? A: To simplify a radical expression, you can factor the numerator and denominator, apply the quotient rule, simplify the radical, apply the product rule, and apply the power rule.

Q: What is the final simplified expression for $\sqrt{\frac{4 a^{10}}{25 b^6}}$? A: The final simplified expression is $\frac{\sqrt{2} a^5 b^{-3/2}}{5}$.