According To The Property A B = A B \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} B ​ A ​ ​ = B A ​ ​ , Which Choice Is Equivalent To The Quotient Below? 125 5 \sqrt{\frac{125}{5}} 5 125 ​ ​ A. 2 B. -5 C. 25 D. 5

Understanding the Property of Square Roots

The property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ is a fundamental concept in mathematics that allows us to simplify expressions involving square roots. This property states that the quotient of two square roots is equal to the square root of the quotient of the numbers inside the square roots.

Applying the Property to the Given Expression

We are given the expression $\sqrt{\frac{125}{5}}$. To simplify this expression, we can apply the property of square roots. First, we can simplify the fraction inside the square root:

$\frac{125}{5} = 25$

Now, we can rewrite the expression as:

$\sqrt{25}$

Simplifying the Square Root

The square root of 25 is equal to 5, since 5 multiplied by 5 equals 25. Therefore, the simplified expression is:

$5$

Comparing the Simplified Expression to the Answer Choices

Now that we have simplified the expression, we can compare it to the answer choices. The correct answer is:

D. 5

The other answer choices, A. 2, B. -5, and C. 25, are not equivalent to the simplified expression.

Conclusion

In conclusion, the property of square roots allows us to simplify expressions involving square roots. By applying this property to the given expression, we can simplify the quotient and arrive at the correct answer. This concept is essential in mathematics and is used extensively in various mathematical operations.

Additional Examples

To further illustrate the concept, let's consider a few more examples:

• $\sqrt{\frac{16}{4}} = \sqrt{4} = 2$
• $\sqrt{\frac{9}{1}} = \sqrt{9} = 3$
• $\sqrt{\frac{36}{9}} = \sqrt{4} = 2$

In each of these examples, we can simplify the expression by applying the property of square roots.

Real-World Applications

The property of square roots has numerous real-world applications. For instance, in physics, the square root of a quantity is often used to represent the magnitude of a vector. In engineering, the square root of a quantity is used to represent the magnitude of a force or a stress.

Final Thoughts

In conclusion, the property of square roots is a fundamental concept in mathematics that allows us to simplify expressions involving square roots. By applying this property, we can simplify the quotient and arrive at the correct answer. This concept is essential in mathematics and is used extensively in various mathematical operations.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Glossary

• Square root: a number that, when multiplied by itself, gives the original number
• Quotient: the result of dividing one number by another
• Property of square roots: a fundamental concept in mathematics that allows us to simplify expressions involving square roots
  Frequently Asked Questions (FAQs) About Square Roots =====================================================

Q: What is a square root?

A: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you can use the property of square roots, which states that the square root of a quotient is equal to the quotient of the square roots. For example, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Q: What is the difference between a square root and a square?

A: A square root is a value that, when multiplied by itself, gives the original number, while a square is the result of multiplying a number by itself. For example, the square root of 16 is 4, while the square of 4 is 16.

Q: Can I simplify a square root expression with a negative number?

A: Yes, you can simplify a square root expression with a negative number. For example, $\sqrt{-16} = \sqrt{-1} \times \sqrt{16} = 4i$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a variable?

A: To simplify a square root expression with a variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{x^2} = x$ or $\sqrt{y^2} = y$, depending on the value of the variable.

Q: Can I simplify a square root expression with a fraction?

A: Yes, you can simplify a square root expression with a fraction. For example, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Q: How do I simplify a square root expression with a decimal?

A: To simplify a square root expression with a decimal, you can use the property of square roots and the rules of algebra. For example, $\sqrt{0.25} = \sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.

Q: Can I simplify a square root expression with a negative decimal?

A: Yes, you can simplify a square root expression with a negative decimal. For example, $\sqrt{-0.25} = \sqrt{-\frac{1}{4}} = \frac{\sqrt{-1}}{\sqrt{4}} = \frac{i}{2}$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a negative variable?

A: To simplify a square root expression with a negative variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{-x^2} = \sqrt{-1} \times \sqrt{x^2} = ix$, where $i$ is the imaginary unit.

Q: Can I simplify a square root expression with a complex number?

A: Yes, you can simplify a square root expression with a complex number. For example, $\sqrt{3+4i} = \sqrt{3} + \sqrt{4i} = \sqrt{3} + 2i$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a radical?

A: To simplify a square root expression with a radical, you can use the property of square roots and the rules of algebra. For example, $\sqrt{a^2} = a$ or $\sqrt{b^2} = b$, depending on the value of the radical.

Q: Can I simplify a square root expression with a mixed number?

A: Yes, you can simplify a square root expression with a mixed number. For example, $\sqrt{2\frac{1}{2}} = \sqrt{\frac{5}{2}} = \frac{\sqrt{5}}{\sqrt{2}}$.

Q: How do I simplify a square root expression with a fraction and a variable?

A: To simplify a square root expression with a fraction and a variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$.

Q: Can I simplify a square root expression with a negative fraction and a variable?

A: Yes, you can simplify a square root expression with a negative fraction and a variable. For example, $\sqrt{-\frac{x}{y}} = \frac{\sqrt{-x}}{\sqrt{y}} = \frac{i\sqrt{x}}{\sqrt{y}}$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a complex fraction and a variable?

A: To simplify a square root expression with a complex fraction and a variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{\frac{a+bi}{c+di}} = \frac{\sqrt{a+bi}}{\sqrt{c+di}}$, where $a$, $b$, $c$, and $d$ are real numbers.

Q: Can I simplify a square root expression with a radical and a variable?

A: Yes, you can simplify a square root expression with a radical and a variable. For example, $\sqrt{a^2x} = ax$ or $\sqrt{b^2y} = by$, depending on the value of the radical and the variable.

Q: How do I simplify a square root expression with a mixed number and a variable?

A: To simplify a square root expression with a mixed number and a variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{2\frac{1}{2}x} = \sqrt{\frac{5}{2}x} = \frac{\sqrt{5x}}{\sqrt{2}}$.

Q: Can I simplify a square root expression with a negative mixed number and a variable?

A: Yes, you can simplify a square root expression with a negative mixed number and a variable. For example, $\sqrt{-2\frac{1}{2}x} = \sqrt{-\frac{5}{2}x} = \frac{i\sqrt{5x}}{\sqrt{2}}$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a complex number and a variable?

A: To simplify a square root expression with a complex number and a variable, you can use the property of square roots and the rules of algebra. For example, $\sqrt{a+bi} = \sqrt{a} + \sqrt{bi} = \sqrt{a} + i\sqrt{b}$, where $a$ and $b$ are real numbers.

Q: Can I simplify a square root expression with a radical and a complex number?

A: Yes, you can simplify a square root expression with a radical and a complex number. For example, $\sqrt{a^2+bi} = a + \sqrt{bi} = a + i\sqrt{b}$, where $a$ and $b$ are real numbers.

Q: How do I simplify a square root expression with a mixed number and a complex number?

A: To simplify a square root expression with a mixed number and a complex number, you can use the property of square roots and the rules of algebra. For example, $\sqrt{2\frac{1}{2}+3i} = \sqrt{\frac{5}{2}+3i} = \frac{\sqrt{5}+3i}{\sqrt{2}}$.

Q: Can I simplify a square root expression with a negative mixed number and a complex number?

A: Yes, you can simplify a square root expression with a negative mixed number and a complex number. For example, $\sqrt{-2\frac{1}{2}+3i} = \sqrt{-\frac{5}{2}+3i} = \frac{i\sqrt{5}+3i}{\sqrt{2}}$, where $i$ is the imaginary unit.

Q: How do I simplify a square root expression with a complex number and a radical?

A: To simplify a square root expression with a complex number and a radical, you can use the property of square roots and the rules of algebra. For example, $\sqrt{a+bi} = \sqrt{a} + \sqrt{bi} = \sqrt{a} + i\sqrt{b}$, where $a$ and $b$ are real numbers.

Q: Can I simplify a square root expression with a radical and a complex number?

A: Yes, you can simplify a square root expression with a radical and a complex number. For example, $\sqrt{a^2+bi} = a + \sqrt{bi} = a + i\sqrt{b}$, where $a$ and $b$ are real numbers.

Q: How do I simplify a square root expression with a mixed number and a radical?

A: To simplify a square