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Calculating n th roots can be done using a similar method, with modifications to deal with n. While computing square roots entirely by hand is tedious. Estimating higher n th roots, even if using a calculator for intermediary steps, is significantly more tedious. For those with an understanding of series, refer here for a more mathematical algorithm for calculating n th roots. For a simpler, but less efficient method, continue to the following steps and example. To calculate n√ a: Fill in the blank spaces in the right quadrant. You must fill each blank space you've just written in the right quadrant with the same integer. This integer must be the largest integer that allows the result of the multiplication problem in the right quadrant to be lower than or equal to the current number on the left. [8] X Research source Let's use the square root of 147 as an example. 147 isn't the product of two perfect squares, so we can't get an exact integer value as above. However, it is the product of one perfect square and another number - 49 and 3. We can use this information to write our answer in simplest terms as follows:

Much as I love crunchy salads, I’m surprised to discover that I prefer the texture of David’s blanched celeriac, which comes closest to my memories of the French kind, being both snappy but slightly yielding. If you’re in a hurry, or like the idea of keeping it completely raw, however, feel free to skip this step. The dressing Repeat step 5 and 6. Find the biggest digit to fill in the blanks on the right that gives an answer lesser than or equal to the current number on the left. Then, solve the problem. [9] X Research source Take the square roots of your perfect square factors. The product property of square roots states that for any given numbers a and b, Sqrt(a × b) = Sqrt(a) × Sqrt(b). Because of this property, we can now take the square roots of our perfect square factors and multiply them together to get our answer. [3] X Research source

Didn’t think you could get salon-level colour from a box? Think again! At Savers, you can get professional-looking results at home with our wide selection of hair dye. And don’t worry – they’re all easy to apply and hard to get wrong! Read more Find the biggest number whose square is less than or equal to S a. The first digit A in our answer is then the biggest integer where the square does not exceed S a (meaning A so that A² ≤ Sa < (A+1)²). In our example, S a = 7, and 2² ≤ 7 < 3², so A = 2. Repeat step 4. Drop the next chunk of the number you're finding the square root of down. When you reach the decimal point in your number, write a decimal point in your answer in the top right quadrant. Then, multiply the number in the top right by 2 and write it next to the blank multiplication problem ("_ × _") as above.

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Sqrt(2 × 2 × 2 × 11). We have several 2's in our square root. Since 2 is a prime number, we can remove a pair and put one outside the square root.

Note: An integer has no fractional or decimal part, and thus a perfect square (which is also an integer) has no fractional or decimal part. Drop down the next pair. Move the next "chunk" in the number whose square root you're solving for down next to the subtracted value you just found. Next multiply the number in the top right quadrant by two and write it in the bottom right quadrant. Next to the number you just wrote down, set aside space for a multiplication problem you'll do in the next step by writing '"_×_="'. As an example, let's find the square root of 45 using this method. We know that 45 = 9 × 5 and we know that 9 = 3 × 3. Thus, we can write our square root in terms of its factors like this: Sqrt(3 × 3 × 5). Simply remove the 3's and put one 3 outside the square root to get your square root in simplest terms: (3)Sqrt(5). From here, it's simple to estimate. David Tanis finishes off his celeriac remoulade with chopped chives and a dusting of cayenne pepper.Estimate, if necessary. With your square root in simplest terms, it's usually fairly easy to get a rough estimate of a numerical answer by guessing the value of any remaining square roots and multiplying through. One way to guide your estimates is to find the perfect squares on either side of the number in your square root. You'll know that the decimal value of the number in your square root is somewhere between these two numbers, so you'll be able to guess in between them. Subtract the number you just calculated from the leftmost pair. As with long division, the next step is to subtract the square we just found from the chunk we just analyzed. Write this number underneath the first chunk and subtract, writing your answer underneath. [7] X Research source Separate your number's digits into pairs. This method uses a process similar to long division to find an exact square root digit-by-digit. Though it's not essential, you may find that it's easiest to perform this process if you visually organize your workspace and your number into workable chunks. First, draw a vertical line separating your work area into two sections, then draw a shorter horizontal line near the top of the right section to divide the right section into a small upper section and a larger lower section. Next, separate your number's digits into pairs, starting from the decimal point. For instance, following this rule, 79,520,789,182.47897 becomes "7 95 20 78 91 82. 47 89 70". Write your number at the top of the left space. [6] X Research source