Dividing Square Roots: A Step-by-Step Guide to Simplifying and Solving Complex Equations


Dividing square roots is a popular mathematical operation involving radicals. If you want to find the quotient of two square roots, you need to know how to rationalize the denominator. This step-by-step process will help you better understand how to divide roots and solve problems like a math pro.

First, let’s take a look at the basic rule for dividing square roots. When you have a fraction with a radical in the denominator, you need to rationalize it. To do this, you multiply both the numerator and the denominator by a factor that will remove the radical from the denominator without changing the value of the fraction.

For example, if you have the expression 46 divided by √45, you can start by rewriting the radical as the square root of a fraction: √(45/1). Now, to rationalize the denominator, you need to find the value that, when multiplied by the bottom, creates a new denominator that is a rational number.

One way to do this is to find a factor that is similar to the original denominator. In this case, the square root of 45 is between 6 and 7. So, you can rewrite the expression as 46 divided by (√(45/1)) times (√(45/45)), which simplifies to 46 divided by (√(45/1)) times √(1/45).

Now that the denominator is rationalized, you can proceed with the division. Divide the numerator (46) by the square root of 45, and divide the denominator (√(45/1)) times √(1/45). This will give you the quotient of the two square roots.

By following this step-by-step approach, you can divide any two square roots and find the value of the quotient. Make sure to practice and test your skills with different examples to better understand the concept of dividing roots.

Remember, dividing square roots is an authorized approach in arithmetic and other fields of math. While the description provided here is a basic explanation, there may be more specific and complex methods for certain problems. If you want to learn more about dividing roots or need further explanation, there are many resources available that can provide additional help and explanation.

In conclusion, dividing square roots involves rationalizing the denominator and following a set of rules. By understanding and applying these rules, you can successfully divide roots and find the quotient.

GRE Math – How to Divide Square Roots

Dividing square roots involves working with fractions and simplifying the quotient step-by-step. It is important to understand how to divide square roots because it is a common concept tested on the GRE Math exam. This article will guide you through the process of dividing square roots in a step-by-step approach.

When dividing square roots, it is important to remember that the denominator should not be zero. A fraction with a denominator of zero is undefined. Also, make sure to work with real numbers, as square roots of negative numbers involve complex numbers and are beyond the scope of this article.

To divide square roots, we start by rewriting the division problem as a fraction. For example, if we want to divide √50 by √2, we can write it as (√50) / (√2).

Next, we simplify the numerator and denominator separately. For the numerator (√50), we can simplify it by factoring out the square root of the largest perfect square, which is 25. So, (√50) = (√25 * √2) = (5√2).

Similarly, for the denominator (√2), we leave it as it is since it cannot be simplified further.

Now, we can rewrite the division problem as (5√2) / (√2).

To simplify this further, we can divide the common factor (√2) in the numerator and denominator, which results in a quotient of 5.

So, (√50) / (√2) = 5.

Here’s another example:

If we want to divide √43 by √45, we can write it as (√43) / (√45).

For the numerator (√43), there are no perfect square factors, so we can’t simplify it further.

For the denominator (√45), we can factor it as (√9 * √5), which equals 3√5.

Now, we can rewrite the division problem as (√43) / (3√5).

We can’t simplify further because there are no common factors in the numerator and denominator.

So, (√43) / (√45) cannot be simplified any further.

To summarize, when dividing square roots, simplify the numerator and denominator separately and rewrite the division problem as a fraction. Then, look for common factors and divide them out. If there are no common factors, the quotient is the final answer.

These steps can be applied to any division problem involving square roots. To learn more and find additional examples, consult a GRE Math prep book or continue to practice using online resources.

Please note that all copyrights and complaints about the content should be authorized and directed to the original source. This article is provided in good faith to assist with understanding the concept of dividing square roots for GRE Math preparation.

Example Question 43 Arithmetic

In this example, we have a division of two square roots. Let’s take a look at how to solve it:

If we want to divide two radicals, we need to rationalize the denominator. In this case, our expression is:

√46 / √2

To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a radical is obtained by changing the sign of the radical’s term. So our expression becomes:

(√46 / √2) * (√2 / √2)

Multiplying the numerators and denominators, we get:

(√46 * √2) / (√2 * √2)

Simplifying the radicals, we have:

(√92) / 2

Since √92 can be simplified to 2√23, the expression becomes:

2√23 / 2

Now, the fraction can be simplified further by canceling out the common factor of 2:

√23 / 1

Therefore, the value of the division is √23.

Remember that dividing radicals follows a specific rule, and you should always rationalize the denominator if necessary. It is also important to note that the result may not be a whole number, as in this case where the quotient is a radical.

Example Question 44 Arithmetic

Let’s take a look at an example that involves division of roots. Suppose we have the expression:

√49 ÷ √44

To simplify this expression, we can take the following steps:

  1. Find the values of the radicands (the numbers under the radical sign): 49 and 44.
  2. Divide the radicands: 49 ÷ 44 = 1.1136 (approximately).
  3. Now, we need to simplify the resulting quotient. Since both radicands are perfect squares, we can rewrite the expression as:


To rationalize the denominator (remove the radical from the bottom), we multiply both the numerator and denominator by the square root of the denominator:

√(49/44) * (√44/√44) = √(49 * 44) / √(44 * 44) = √2156 / 44

This expression cannot be simplified further, as the radicand (2156) does not have any perfect squares as factors. Therefore, the quotient of the given expression is √2156 / 44.

Remember, when dividing radicals, you should always try to simplify the radicands first and then rationalize the denominator if necessary. This approach ensures that you find the most simplified form of the expression.

Example Question 45 Arithmetic

To divide roots, we can use the rule that states “the quotient of two square roots is equal to the square root of the quotient”. In other words, if we have √a/√b, we can rewrite it as √(a/b). This approach is useful when working with radicals, as it simplifies the arithmetic involved.

Let’s take a look at an example question:

Question: Simplify √49/√43

Step-by-Step Explanation:

  1. First, we can rewrite the expression as √(49/43).
  2. Next, we divide 49 by 43. The quotient is approximately 1.1395.
  3. Therefore, the simplified form of √49/√43 is √1.1395.

It is important to note that when dividing radicals, we should always simplify the expression as much as possible. In this case, we divided the radicand (the number inside the radical) in order to get a rational number. However, sometimes it may not be possible to divide the radicand evenly. In such cases, we can leave the expression as it is, without further simplification.

If you want to learn more about dividing roots and other arithmetic techniques, there are many resources available online. Some popular websites for learning arithmetic include Khan Academy, MathisFun, and Purplemath. Additionally, there are authorized test prep resources for exams like the GRE and DMCA that provide step-by-step explanations and practice problems.

Remember, dividing roots is just one aspect of arithmetic. It is important to have a good understanding of pre-algebra and basic arithmetic operations before diving into more complex topics like dividing radicals. With practice and the right resources, anyone can improve their arithmetic skills and solve arithmetic problems with ease.

Example Question 46: Arithmetic

Let’s take a look at a sample question:

A person is authorized to read confidential documents, but is unsure of the specific value of a certain number. The person knows that the sum of the square roots of two numbers is 44, and the difference of the square roots is 2. What is the value of the number?

To solve this problem, we need to divide the radicals and use the rules of arithmetic:

  1. First, we rewrite the given information in terms of radicals:
  • The sum of the square roots is $\sqrt{a} + \sqrt{b} = 44$
  • The difference of the square roots is $\sqrt{a} – \sqrt{b} = 2$
  • Next, we multiply the two equations together to eliminate the radicals:
    • $(\sqrt{a} + \sqrt{b})(\sqrt{a} – \sqrt{b}) = 44 \cdot 2$
    • $(\sqrt{a})^2 – (\sqrt{b})^2 = 88$ (using the difference of squares rule)
    • $a – b = 88$
  • Now, we have a system of equations. We can solve the system by adding the two equations together:
    • $a + b = 44$
    • $a – b = 88$
    • $2a = 132$
    • $a = 66$
  • Finally, we can substitute the value of $a$ back into one of the original equations to solve for $b$:
    • $66 + b = 44$
    • $b = -22$

    So the value of the number in question is -22.

    This example demonstrates the process of dividing square roots to solve arithmetic problems. It’s important to know the rules of arithmetic and how to simplify radicals to better understand and solve problems involving roots.

    If you want more examples and a detailed explanation of how to divide roots, you can find it in the GRE arithmetic section of your test prep materials.

    This article is just a sample to help you understand the approach and steps involved in dividing roots. Make sure to read the specific information provided in your test preparation materials.

    For more information about arithmetic, fractions, and rationalizing the denominator, please read our DMCA policy.

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    Dr Heidi Parkes

    By Dr Heidi Parkes

    Senior Information Extension Officer QLD Dept of Agriculture & Fisheries.