Calculate a Cube Root Using Linear Approximation

By Mark Ryan

Linear approximation is not only easy to do, but also very useful! For example, you can use it to approximate a cubed root without using a calculator.

Here’s an example. Can you approximate

Cubed root of 70

in your head? Yes, you can! How?

Like this: Bingo! 4.125.

Well, okay, there’s a little more to it than that. Take a look at the figure and then follow the steps below to get the full picture.

The line tangent to the curve at (64, 4) can be used to approximate cube roots or numbers near 64.
The line tangent to the curve at (64, 4) can be used to approximate cube roots or numbers near 64.

To estimate

The cube root for 70

follow these steps:

  1. Find a perfect cube root near

    The cube root of 70

    You notice that

    Symbol for the cubed root of 70

    is near a no-brainer,

    The cubed root of 64

    which, of course, is 4. That gives you the point (64, 4) on the graph of

    Y equals the cubed root of x

  2. Find the slope of

    The equation y equals the cube root for x.

    (which is the slope of the tangent line) at x = 64.

    The slope of a tangent line of 64

    This tells you that — to approximate cube roots near 64 — you add (or subtract)

    A forty-eighth.

    to 4 for each increase (or decrease) of one from 64. For example, the cube root of 65 is about

    Four and a forty-eighth.

    the cube root of 66 is about

    Four and two forty-eights or four and a twenty-fourth

    the cube root of 67 is about

    Four and three fourty-eighths or four and a sixteenth.

    and the cube root of 63 is about

    three and forty-seven fourty-eighths.

  3. Use the point-slope form to write the equation of the tangent line at (64, 4).

    The point-slope formula for the tangent line at (64, 4)

    In the third line of the above equation, you put the 4 in the front of the right side of the equation (instead of at the far right which might seem more natural) for two reasons. First, because doing so makes this equation jibe with the explanation at the end of Step 2 about starting at 4 and going up (or down) from there as you move away from the point of tangency. And second, to make this equation agree with the explanation at the end of Step 4. You’ll see how it all works in a minute.

  4. Because this tangent line runs so close to the function

    The function y equals the cubed root for x

    near x = 64, you can use it to estimate cube roots of numbers near 64, like at x = 70.

    Using the slope form to find the cubed root for 70

    By the way, in your calc text, the simple point-slope form from algebra (first equation line in Step 3) is probably rewritten in highfalutin’ calculus terms — like this:

    The point-slope form in calculus terms.

Don’t be intimidated by this equation. It’s just your friendly old algebra equation in disguise! Look carefully at it term by term and you’ll see that it’s mathematically identical to the point-slope equation tweaked like this: y = y1 + m(xx1). (The different subscript numbers, 0 and 1, have no significance.)