Hey, this is Presh Talwalkar reminding you to simplify your choices and prove your ### answer

to the following expression. You may not use calculators or computers. I've received many email requests for this type of issue. I think Akshat in India because you suggested it to me Can you figure it out first try this problem and when you are ready watch the video to find a solution. How can we simplify this expression? I first write a as the number under the first cube root and then let b be a conjugate square root under the second cube root. Notice that the cube root of a is a real number and the cube root of b is a real number, so their sum will also be a real number, so we end up looking for a real number a plus the cube root of b.

We equate this to x. Now we roll both sides and then we expand. Now let's simplify this expression. We're going to put the squares in these cube roots first. Now let's group these cube roots together. We can then write a squared as a and b squared as bb. Notice that we have an a b term here and a ba term here and they are equal to each other which is the product of square root conjugates The nice thing is that it will be a difference of squares and this simplifies to to be minus five to the power of three.

We replace that with and now minus five to the third power, if we take the cube root of that we get minus 5. 3 times minus 5 equals minus 15. so then we have minus 15 and we can group the remaining terms that are the cube root of a plus the cube root of b will be. We now recall that x was defined to be exactly that quantity so we can substitute. Now a plus b equals 16 because the square root terms cancel, so we end up with a simple polynomial expression: x to the power of three equals 16 minus 15x if there is a solution in this equation for which we are looking for a real value is the ### answer

to our original expression, so we write x to the power of three plus 15x minus 16 equals 0.

How can we solve this equation well? Usually you can try specific values and you will quickly see that x equals 1 is a solution to this equation. So x minus 1 is a factor of this polynomial. We can factor this into the following product of x minus 1 and squared x squared plus x plus 16. We can solve the square using the quadratic formula, but we will learn that the square roots both involve non-zero imaginary parts, so they're extraneous roots can't be the ### answer

to our original problem because we're looking for a real value of x Looking for. This means that the ### answer

x is one.

Double check this exercise by entering it into Wolfram Alpha. If she gives the ### answer

of one, you might be surprised to find that she gives a complex ### answer

with a non-zero imaginary part. What's happening? Wolfram Alpha uses the main root. The reason for this is the equation x to the power of three equals c has three solutions and Wolfram Alpha takes the principal branch cut which is #### most

commonly used in higher math but we don't want the principal branch cut we want the really valuable root so how can we Forcing Wolfram Alpha that there are a few of these.

One is that you can use the command of cube root which actually gives you the root really valued and you get the ### answer

one and for other roots you can use the absurd command of x use comma. This gives you the real nth root of x It's a pretty neat problem. Thanks for watching these free math videos available on YouTube. They boost student confidence and inspire viewers around the world to make mathematical discoveries. They have over 100 million views and the channel has over 1 million subscribers. Please subscribe to it for free Get the latest videos. You can email me a puzzle or math topic. presh mindyourdecisions.com.

If you like you can check out my articles on Teespring ion and you can support me on patreon thanks for watching and thanks for your support

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