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How to Estimate Higher-Order Roots

Instructor: Elizabeth Popelka-Brown
In this lesson, students will learn how to calculate and estimate the value of roots. The roots included in specific examples include square, cube, fourth, and fifth, but higher-order roots are mentioned. The examples also include negative roots and what to do when the root is not a whole number or integer and must be estimated.

Imagine you have 25 floor tiles you'd like to arrange into a square. Through trial and error, you discover you can arrange them into a square that is five tiles long and five tiles wide.

square tiles
root,

Squaring a Root

In this situation, you were figuring out what number times itself would give you 25, so that you knew how long and wide the square needed to be. When you multiply 5 by itself, this is called squaring the number 5; when doing the opposite and determining what number is multiplied twice to get 25, you are finding the square root. The tile situation could have been represented in symbolic form with a root symbol, called a radical.


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because


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It should also be noted that 25 has a second square root, -5, since -5 multiplied by -5 is equal to 25, although it doesn't always make sense in real-life situations to have negative roots.

Positive and Negative Cube Roots

Multiplying the same number three times is called cubing. This can be represented by the idea that a cube with a length, width, and height of 5 would be made up of 125 cubic units. When you cube 5, you multiply 5 times 5 times 5 to get 125. The small three, located next to the 5, is the exponent, a shorthand way of stating that the 5 is multiplied 3 times.


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If you do the opposite of cubing, finding the cube root, you are determining what was multiplied three times to arrive at a certain number. For example, the cube root of 216 would be written symbolically as


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The index, in this case, the 3 above, tells how many times the root number must be multiplied. You do NOT multiply 216 three times to find the cube root. The cube root of 216 is 6, because


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There are also negative cube roots. The cube root of -343 would be written symbolically as


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and has a value of -7, because


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Higher Order Roots

Just as there are square and cube roots, there are higher-order roots, roots with a higher index. A fourth root, for example, is the number you multiply 4 times to arrive at another number. The symbol is written like the square and cube roots, but with an index of 4. In this case, there are two roots, as you can see.

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because

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And also

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because

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There are also fifth roots

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And 6th roots

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And higher.

Strategies for Finding a Cube Root

Having your multiplication tables memorized can help with finding a square root, as long as the number is not too big. For example, many of us can determine


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because we know that 12 multiplied by 12 is 144 and that -12 multiplied by -12 is also 144. However, many of us start to struggle when determining cube roots, because we haven't memorized the product of three numbers.

One strategy you can use is simple guessing and checking. An example is shown:


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so


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Another strategy is to factor, break apart into smaller pieces, by multiplication, the number under the radical sign. For example:


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Since 8, broken down into factors, shows 2 multiplied 3 times, the cube root of 8 is 2.

Strategies for Higher Order Roots

Here's another example:

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Since 3 is a factor of 243, repeated 5 times, it is the fifth root of 243.

Estimating Roots

While many numbers have roots that are whole numbers or integers, in real life, there are many more that do not. In these cases, it helps to be able to estimate roots, when an exact value is not needed. Take the following example:

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Since 2*2*2 is 8 and 3*3*3 is 27, the cube root of 17 must be a number between 2 and 3, thus not a whole number.

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