Common Tricks to Simplify Square Roots Faster
In mathematics, squares and square roots are inverse operations. The Square of a number is actually the value of power 2 of the number. Whereas, the square root of a number is the number that we require to multiply by itself to obtain the original number. For simplifying square roots with variables, if m is the square root of n, it implies that m × m = n. The square of any number is always a positive number, thus every number contains two square roots, one positive value, and one negative value.
Simplifying Square Roots – Methods and Examples
As already discussed, the square root is an inverse operation of squaring a number. The square root of a number m is represented using a radical sign √x or x ½. A square root of a number m is such that, a number n is the square of m, simply written as n² = m.
For example, the square root of 49 is denoted as √49 = 7. A number whose square root is computed is called a radicand. In this expression, √49 = 7, number 49 is the radicand.
There are different techniques of simplifying square roots with variables or complex expressions, based upon the number of radicals and the values under each radical.
Methods to Simplify Square Roots
Simplification of the square root of a number entails different methods that are as follows:
Prime Factorization Method
Square Root by Prime factorization of any number means to denote the number as a product of prime numbers. To simplify the square root of a given number by prime factorization method, you need to follow the steps below:
Step 1: Divide the given number into its prime factors.
Step 2: Form pairs of similar factors in a way both factors are equal to each pair.
Step 3: Take out one factor from the pair.
Step 4: Determine the product of the factors obtained by taking one factor from each pair.
Step 5: That product is the square root of the given number.
Let’s find the square root of 144 by prime factorization method:
144 = 2 × 2 × 2 × 2 × 3 × 3
Now pairing the factors;
(2 × 2) × (2 × 2) × (3 × 3)
= 2² × 2² × 3²
= (2 × 2 × 3)²
= 12²
144 = 12²
√144 = 12
Repeated Subtraction Method
The Repeated Subtraction Method of Square Root is very simple. You will require subtracting the consecutive odd numbers from the number for which we are calculating the square root, till you reach 0. The number of times you subtract is the square root of the given number; however, it works only for perfect square numbers. Let us see how the square root of 16 is simplified using this method:
16 – 1 = 15
15 – 3 = 12
12 – 5 = 7
7 – 7 = 0
You can see that we have subtracted 4 times. Therefore, √16 = 4.
Estimation Method
Square Root by estimation indicates a reasonable approximation of the actual value in order to make calculations simpler and realistic. This technique helps in guessing the square root of a given number. Let’s apply this method to find √15. To find the nearest perfect square number to 15, we are familiar with √16 = 4 and √9 = 3. This states that √15 lies between 3 and 4. Now, we need to check if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Thus, √15 lies between 3.5 and 4 and is more close to 4.
Now, consider squares of 3.8 and 3.9. 3.82 = 14.44 and 3.92 = 15.21. This states that √15 lies between 3.8 and 3.9. repeating the process, we see 3.85 is the nearest ant thus √15 = 3.872.
Long Division Method
Square Root by Long Division Method is a technique for dividing large numbers into parts, splitting up the division problem into an order of easier steps. We can find the exact square root of any given number using this method. Let's find and simplify the square root of 180 using the long division method.
Step 1: Place a bar over every pair of the number beginning from the unit’s place (right-most side). You will have two pairs, i.e. 1 and 80.
Step 2: We divide the left-most number by the largest number whose square is ≤ the number in the left-most pair.
Step 3: get the number under the next bar to the right of the remainder. Add up the last digit of the quotient to the divisor. To the right of the sum attained, find an appropriate number which, together with the outcome of the sum, creates a new divisor for the new dividend that is carried down.
Step 4: The new number in the quotient will consist of the same number as chosen in the divisor. The condition is similar— as being either ≤ the dividend.
Step 5: Now, we will use a decimal point and add zeros in pairs to the remainder.
Step 6: The quotient therefore attained is the square root of the number.
Fun Facts
Simplify square roots using the prime factorization method works when the given number is a perfect square number.
FAQs on How to Simplify Square Roots: Step-by-Step Guide
1. What are the basic steps to simplify a square root using the prime factorization method?
To simplify a square root, follow these key steps:
1. Find the prime factors of the number inside the square root sign (the radicand).
2. Look for pairs of identical factors.
3. For each pair of factors, take one of those factors out from under the square root sign.
4. Multiply all the factors you pulled out.
5. Any factors that did not have a pair remain inside the square root sign. The simplified form is the product of the numbers outside and the remaining number inside the root.
2. Can you provide an example of simplifying a square root, such as √50?
Yes. To simplify √50, first find the prime factors of 50, which are 2 × 5 × 5. Here, we have a pair of 5s. We can take one 5 out of the square root. The number 2 does not have a pair, so it stays inside. Therefore, the simplified form of √50 is 5√2.
3. What is a 'perfect square' and why is it important for simplifying radicals?
A perfect square is a number that is the product of an integer with itself. For example, 25 is a perfect square because 5 × 5 = 25. Perfect squares are crucial for simplifying radicals because the square root of a perfect square is a whole number (e.g., √25 = 5). The quickest way to simplify a radical is often to find the largest perfect square factor of the number and then take its square root out of the radical sign.
4. How do you simplify a more complex square root, like √72?
To simplify √72, we look for the largest perfect square that divides 72. The perfect squares are 4, 9, 16, 25, 36, etc. We see that 36 is a factor of 72 (72 = 36 × 2). We can rewrite √72 as √(36 × 2). Since the square root of 36 is 6, we can pull the 6 outside the radical, leaving the 2 inside. Thus, the simplified form of √72 is 6√2.
5. What is the main goal when a question asks to 'simplify' a square root?
The main goal of simplifying a square root is to express it in its most 'basic' form. This doesn't mean finding a decimal approximation. Instead, it means rewriting the radical so that the number under the radical sign (the radicand) has no perfect square factors other than 1. This gives an exact value (like 5√2) which is more precise and useful in algebra than a long decimal (like 7.071...).
6. Why can't a square root with a prime number, like √17, be simplified?
A square root like √17 cannot be simplified because the process of simplification depends on finding perfect square factors within the number. A prime number, by definition, only has two factors: 1 and itself. Since 17 has no perfect square factors other than 1, it is already in its simplest possible form. There are no pairs of factors to pull out from under the radical sign.
7. How does the simplification process change when variables are involved, for example, √(x⁵)?
The principle remains the same: look for pairs. For variables with exponents, a 'pair' corresponds to an even power. You can break down √(x⁵) into √(x⁴ ⋅ x). The square root of x⁴ is x² (because x² ⋅ x² = x⁴). So, you can pull x² out of the radical. The single 'x' remains inside. Therefore, the simplified form of √(x⁵) is x²√x.
8. What is the difference between simplifying a square root and finding its decimal value?
Simplifying a square root provides an exact mathematical expression, while finding its decimal value usually results in an approximation. For example:
- Simplifying √50 gives 5√2. This is an exact value.
- The decimal value of √50 is approximately 7.07106..., which is rounded.
