Square Root Of 0.09: A Step-by-Step Solution

The question at hand asks us to determine the value of the square root of 0.09. This seemingly simple problem delves into the fundamental concepts of square roots, decimal representation, and fractional equivalence. To effectively tackle this, we must break down the problem into manageable steps, ensuring a clear understanding of each stage. The square root of a number is a value that, when multiplied by itself, equals the original number. In mathematical notation, the square root of a number 'x' is represented as √x. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9 (3 * 3 = 9). Similarly, we need to find a number that, when multiplied by itself, results in 0.09. Decimals, as we know, are a way of representing numbers that are not whole numbers. They consist of a whole number part and a fractional part, separated by a decimal point. The digits to the right of the decimal point represent fractions with denominators that are powers of 10. For example, 0.1 represents one-tenth (1/10), 0.01 represents one-hundredth (1/100), and 0.001 represents one-thousandth (1/1000). Converting decimals to fractions is a crucial skill in mathematics, allowing us to manipulate and simplify expressions more easily. In this particular case, 0.09 can be expressed as the fraction 9/100. This conversion is achieved by recognizing that 0.09 has two digits after the decimal point, indicating that it represents nine hundredths. This conversion now sets the stage for simplifying the square root operation, paving the way for a clearer path to the solution. By understanding the principles of square roots and decimal-to-fraction conversions, we can approach this problem with a solid foundation.

Decimal to Fraction Conversion

Before diving directly into the square root calculation, converting the decimal 0.09 into its equivalent fractional form is a crucial step. This conversion simplifies the process and makes it easier to apply the properties of square roots. Understanding how decimals relate to fractions is fundamental in mathematics, and this step reinforces that understanding. As mentioned earlier, decimals represent fractions with denominators that are powers of 10. The number of digits after the decimal point indicates the power of 10 in the denominator. In the case of 0.09, there are two digits after the decimal point, which means the denominator will be 10 raised to the power of 2, or 100. Therefore, 0.09 can be written as 9/100. This representation clearly shows that 0.09 is equivalent to nine hundredths. This conversion allows us to rewrite the original problem, which asked for the square root of 0.09, as finding the square root of 9/100. This transformation is significant because it allows us to apply a key property of square roots: the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator. In mathematical terms, √(a/b) = √a / √b. This property simplifies the calculation by breaking down the problem into two smaller, more manageable square root operations. We now need to find the square root of 9 and the square root of 100 separately. The square root of 9 is 3, because 3 multiplied by itself (3 * 3) equals 9. Similarly, the square root of 100 is 10, because 10 multiplied by itself (10 * 10) equals 100. With these individual square roots determined, we can now combine them to find the square root of the entire fraction. The square root of 9/100 is therefore 3/10. This fraction represents the solution in its simplest form, but we can also convert it back to a decimal to verify our answer and compare it with the original options. Converting 3/10 to a decimal involves dividing 3 by 10, which results in 0.3. This decimal representation further confirms that the square root of 0.09 is indeed 0.3. The process of converting decimals to fractions and vice versa is a vital skill in mathematics. It not only simplifies calculations but also provides a deeper understanding of the relationship between these two forms of representing numbers.

Calculating the Square Root

Now that we've converted 0.09 into the fraction 9/100, we can proceed to calculate the square root. As established earlier, the square root of a fraction is the square root of the numerator divided by the square root of the denominator. This crucial property allows us to break down the problem into simpler components. We need to find √9 and √100 separately. The square root of 9 is the number that, when multiplied by itself, equals 9. We know that 3 * 3 = 9, therefore, √9 = 3. The square root of 100 is the number that, when multiplied by itself, equals 100. We know that 10 * 10 = 100, therefore, √100 = 10. Now that we have the square roots of the numerator and the denominator, we can combine them to find the square root of the entire fraction. √ (9/100) = √9 / √100 = 3/10. This result, 3/10, represents the square root of 0.09 in fractional form. To further understand and verify our result, we can convert this fraction back into a decimal. Dividing 3 by 10 gives us 0.3. We can check if 0.3 is indeed the square root of 0.09 by multiplying 0.3 by itself: 0.3 * 0.3 = 0.09. This confirms that our calculation is correct. The square root of 0.09 is indeed 0.3. This process of calculating square roots, especially for fractions and decimals, highlights the importance of understanding the fundamental properties of these mathematical operations. It also emphasizes the connection between different forms of number representation, such as fractions and decimals. Being able to move fluently between these forms is a key skill in mathematics and problem-solving. Furthermore, this example demonstrates how breaking down a complex problem into smaller, more manageable steps can make the solution process clearer and more accessible. By understanding the relationship between square roots, fractions, and decimals, we can confidently tackle similar problems in the future.

Converting the Result to Decimal Form

After successfully calculating the square root of 9/100 as 3/10, the next step is to convert this fraction back into decimal form. This conversion is essential for comparing our result with the answer choices provided, which are likely in either decimal or fractional form. Furthermore, converting between fractions and decimals reinforces the understanding of their equivalence and provides a practical application of this knowledge. A fraction represents a part of a whole, and a decimal is another way of expressing this part using a base-10 system. To convert a fraction to a decimal, we simply divide the numerator (the top number) by the denominator (the bottom number). In this case, we need to divide 3 by 10. The division of 3 by 10 results in 0.3. This can be understood intuitively by recognizing that 3/10 represents three-tenths, which is directly expressed as 0.3 in decimal form. The decimal 0.3 has one digit after the decimal point, which corresponds to the tenths place. This conversion confirms that the fractional representation of the square root, 3/10, is equivalent to the decimal representation 0.3. Now, to ensure our answer is correct, we can verify that 0.3 is indeed the square root of 0.09 by multiplying 0.3 by itself. 0.3 multiplied by 0.3 equals 0.09 (0.3 * 0.3 = 0.09). This calculation confirms that 0.3 is the correct square root of 0.09. The ability to convert fractions to decimals and vice versa is a fundamental skill in mathematics, particularly when dealing with square roots, percentages, and other numerical calculations. This skill not only helps in solving problems but also provides a deeper understanding of the relationship between different numerical representations. In this case, the conversion from 3/10 to 0.3 allowed us to confidently identify the correct answer and verify its accuracy. Understanding the equivalence between fractions and decimals enhances our mathematical fluency and problem-solving abilities.

Identifying the Correct Option

Having determined that the square root of 0.09 is 0.3, and having converted this result to both fractional (3/10) and decimal forms, we are now equipped to identify the correct option from the given choices. The options presented are:

A. 3/10 B. 1/9 C. 1/3 D. 3/100

Our calculated result, 3/10, directly corresponds to option A. Therefore, option A is the correct answer. To further solidify our understanding, let's examine why the other options are incorrect. Option B, 1/9, is a fraction that represents one-ninth. To convert this to a decimal, we would divide 1 by 9, which results in approximately 0.1111. This value is significantly different from 0.3, so option B is incorrect. Option C, 1/3, is a fraction that represents one-third. Converting this to a decimal involves dividing 1 by 3, which results in approximately 0.3333. While this value is closer to 0.3 than option B, it is still not equal to 0.3, making option C incorrect. Option D, 3/100, is a fraction that represents three-hundredths. Converting this to a decimal involves dividing 3 by 100, which results in 0.03. This value is significantly smaller than 0.3, so option D is also incorrect. This process of elimination reinforces the accuracy of our calculation and highlights the importance of understanding the values represented by different fractions and decimals. By systematically analyzing each option and comparing it with our calculated result, we can confidently confirm that option A, 3/10, is the correct answer. This exercise also demonstrates the value of converting numbers between different forms (fractions and decimals) to aid in problem-solving and verification. Understanding the relationships between different mathematical representations allows us to approach problems from multiple angles and ensure the accuracy of our solutions. The key to successfully solving mathematical problems often lies in a combination of accurate calculations and a thorough understanding of the underlying concepts.

The final answer is A. 3/10.