Understanding the Sign of the Answer

Oct 13, 2025 by miodeya.com 132 views

Understanding the Problem

The given problem involves finding the square root of a fraction, which is $\frac{900}{196}$ . To simplify this expression, we need to find the prime factorization of both the numerator and the denominator. The numerator, 900, can be factored as $2^2 \times 3^2 \times 5^2$ , while the denominator, 196, can be factored as $2^2 \times 7^2$ .

Simplifying the Fraction

To simplify the fraction, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{900}{196} = \frac{2^2 \times 3^2 \times 5^2}{2^2 \times 7^2} = \frac{3^2 \times 5^2}{7^2}

Finding the Square Root

Now that we have simplified the fraction, we can find the square root of the numerator and the denominator separately. The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Writing the Answer

Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}

Conclusion

In conclusion, the square root of $\frac{900}{196}$ is $\pm \frac{15}{7}$ . This is a proper fraction, which is a fraction where the numerator is less than the denominator.

Final Answer

The final answer is $\boxed{\pm \frac{15}{7}}$ .

Understanding the Sign of the Answer

The given problem involves finding the square root of a fraction, which is $\frac{900}{196}$ . Since the square root of a number can be positive or negative, we need to consider both possibilities. The square root of a positive number is always positive, while the square root of a negative number is always negative.

Simplifying the Fraction with Negative Numbers

To simplify the fraction with negative numbers, we can multiply both the numerator and the denominator by the square root of the negative number. In this case, we can multiply both the numerator and the denominator by $\sqrt{-1}$ , which is equal to $i$ .

Finding the Square Root of a Negative Number

The square root of a negative number is always an imaginary number, which is a complex number that has a non-zero imaginary part. In this case, the square root of $-1$ is $i$ , which is an imaginary unit.

Writing the Answer with Negative Numbers

Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction with negative numbers as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}i

Conclusion with Negative Numbers

In conclusion, the square root of $\frac{900}{196}$ with negative numbers is $\pm \frac{15}{7}i$ . This is an imaginary number, which is a complex number that has a non-zero imaginary part.

Final Answer with Negative Numbers

The final answer with negative numbers is $\boxed{\pm \frac{15}{7}i}$ .

Understanding the Relationship Between the Square Root and the Original Fraction

The square root of a fraction is related to the original fraction in a specific way. The square root of a fraction can be written as the square root of the numerator divided by the square root of the denominator.

Simplifying the Fraction with the Square Root

To simplify the fraction with the square root, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{\sqrt{900}}{\sqrt{196}} = \frac{\sqrt{2^2 \times 3^2 \times 5^2}}{\sqrt{2^2 \times 7^2}} = \frac{3 \times 5}{7}

Finding the Square Root of the Numerator and the Denominator

Now that we have simplified the fraction with the square root, we can find the square root of the numerator and the denominator separately. The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Writing the Answer with the Square Root

Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}

Conclusion with the Square Root

In conclusion, the square root of $\frac{900}{196}$ is $\pm \frac{15}{7}$ . This is a proper fraction, which is a fraction where the numerator is less than the denominator.

Final Answer with the Square Root

The final answer with the square root is $\boxed{\pm \frac{15}{7}}$ .

Understanding the Relationship Between the Square Root and the Original Fraction with Negative Numbers

The square root of a fraction with negative numbers is related to the original fraction in a specific way. The square root of a fraction with negative numbers can be written as the square root of the numerator divided by the square root of the denominator.

Simplifying the Fraction with the Square Root and Negative Numbers

To simplify the fraction with the square root and negative numbers, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{\sqrt{900}}{\sqrt{196}} = \frac{\sqrt{2^2 \times 3^2 \times 5^2}}{\sqrt{2^2 \times 7^2}} = \frac{3 \times 5}{7}

Finding the Square Root of the Numerator and the Denominator with Negative Numbers

Now that we have simplified the fraction with the square root and negative numbers, we can find the square root of the numerator and the denominator separately. The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Writing the Answer with the Square Root and Negative Numbers

Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction with negative numbers as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}i

Conclusion with the Square Root and Negative Numbers

In conclusion, the square root of $\frac{900}{196}$ with negative numbers is $\pm \frac{15}{7}i$ . This is an imaginary number, which is a complex number that has a non-zero imaginary part.

Final Answer with the Square Root and Negative Numbers

The final answer with the square root and negative numbers is $\boxed{\pm \frac{15}{7}i}$ .

Understanding the Relationship Between the Square Root and the Original Fraction with Different Signs

The square root of a fraction with different signs is related to the original fraction in a specific way. The square root of a fraction with different signs can be written as the square root of the numerator divided by the square root of the denominator.

Simplifying the Fraction with the Square Root and Different Signs

To simplify the fraction with the square root and different signs, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{\sqrt{900}}{\sqrt{196}} = \frac{\sqrt{2^2 \times 3^2 \times 5^2}}{\sqrt{2^2 \times 7^2}} = \frac{3 \times 5}{7}

Finding the Square Root of the Numerator and the Denominator with Different Signs

Now that we have simplified the fraction with the square root and different signs, we can find the square root of the numerator and the denominator separately. The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Writing the Answer with the Square Root and Different Signs

Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction with different signs as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}

Conclusion with the Square Root and Different Signs

In conclusion, the square root of $\frac{900}{196}$ with different signs is $\pm \frac{15}{7}$ . This is a proper fraction, which is a fraction where the numerator is less than the denominator.

Final Answer with the Square Root and Different Signs

The final answer with the square root and different signs is $\boxed{\pm \frac{15

Q: What is the square root of $\frac{900}{196}$ ?

A: The square root of $\frac{900}{196}$ is $\pm \frac{15}{7}$ .

Q: How do you simplify the fraction $\frac{900}{196}$ ?

A: To simplify the fraction $\frac{900}{196}$ , we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{900}{196} = \frac{2^2 \times 3^2 \times 5^2}{2^2 \times 7^2} = \frac{3^2 \times 5^2}{7^2}

Q: What is the square root of the numerator and the denominator?

A: The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Q: How do you write the answer as a proper fraction, improper fraction, or whole number?

A: Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}

Q: What is the relationship between the square root and the original fraction?

A: The square root of a fraction is related to the original fraction in a specific way. The square root of a fraction can be written as the square root of the numerator divided by the square root of the denominator.

Q: How do you simplify the fraction with the square root and negative numbers?

A: To simplify the fraction with the square root and negative numbers, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{\sqrt{900}}{\sqrt{196}} = \frac{\sqrt{2^2 \times 3^2 \times 5^2}}{\sqrt{2^2 \times 7^2}} = \frac{3 \times 5}{7}

Q: What is the square root of the numerator and the denominator with negative numbers?

A: The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Q: How do you write the answer with the square root and negative numbers?

A: Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction with negative numbers as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}i

Q: What is the relationship between the square root and the original fraction with negative numbers?

A: The square root of a fraction with negative numbers is related to the original fraction in a specific way. The square root of a fraction with negative numbers can be written as the square root of the numerator divided by the square root of the denominator.

Q: How do you simplify the fraction with the square root and different signs?

A: To simplify the fraction with the square root and different signs, we can cancel out common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of $2^2$ . We can cancel out these common factors to get:

\frac{\sqrt{900}}{\sqrt{196}} = \frac{\sqrt{2^2 \times 3^2 \times 5^2}}{\sqrt{2^2 \times 7^2}} = \frac{3 \times 5}{7}

Q: What is the square root of the numerator and the denominator with different signs?

A: The square root of $3^2 \times 5^2$ is $3 \times 5 = 15$ , while the square root of $7^2$ is 7.

Q: How do you write the answer with the square root and different signs?

A: Since we are asked to write the answer as a proper fraction, improper fraction, or whole number, we can write the square root of the fraction with different signs as:

\pm \sqrt{\frac{900}{196}} = \pm \frac{15}{7}

Q: What is the relationship between the square root and the original fraction with different signs?

A: The square root of a fraction with different signs is related to the original fraction in a specific way. The square root of a fraction with different signs can be written as the square root of the numerator divided by the square root of the denominator.

Q: What is the final answer?

A: The final answer is $\boxed{\pm \frac{15}{7}}$ .

Q: What is the final answer with negative numbers?

A: The final answer with negative numbers is $\boxed{\pm \frac{15}{7}i}$ .

Q: What is the final answer with different signs?

A: The final answer with different signs is $\boxed{\pm \frac{15}{7}}$ .