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It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots. Example: what is the cube root of 30? Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.
There is a fun method for calculating a square root that gets more and more accurate each time around: a) start with a guess (let's guess 4 is the square root of 10) b) divide by the guess (10/4 = 2.5)
The cube root of a number is a special value that, when used in a multiplication three times, gives that number. Example: The cube root of 27 is 3 because 3 × 3 × 3 = 27. Also the cube root of 64 is 4 because 4 × 4 × 4 = 64, and so on.
Exponents, Roots (such as square roots, cube roots etc) and Logarithms are all related! Let's start with the simple example of 3 × 3 = 9:
"Rationalizing the denominator" is when we move a root (like a square root or cube root) from the bottom of a fraction to the top. Oh No! An Irrational Denominator! The bottom of a fraction is called the denominator. Numbers like 2 and 3 are rational. But many roots, such as √2 and √3, are irrational.
A square root can also be written as a fractional exponent of one-half: √ x = x ½ but only for x greater than or equal to 0. How About the Square Root of Negatives? The result is an Imaginary Number... read that page to learn more.
We can get rid of a square root by squaring (or cube roots by cubing, etc). Warning: this can sometimes create "solutions" which don't actually work when we put them into the original equation. So we need to Check!
Illustrated definition of Radical: A square root, cube root, etc. The symbol is radic
Roots of a Polynomial. A "root" (or "zero") is where the polynomial is equal to zero: Put simply: a root is the x-value where the y-value equals zero. General Polynomial. If we have a general polynomial like this: f(x) = ax n + bx n-1 + cx n-2 + ... + z. Then:
Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead he proved the square root of 2 could not be written as a fraction, so it is irrational.