A Brutal Radical Expression Made Easy! The Shortcut Changes Everything.

追加: 2026-06-03

[00:00:02]Good morning math. Evaluate the sum of the fourth power of the quantity square<unk> of 5 +<unk> 3 -<unk> 2 /<unk> 2 and the 4th power of the quantity square<unk> of 5 -<unk> 3 +<unk> 2 /<unk> 2.

[00:00:26]Now let's get started. This time we have a numerical calculation problem. One possible approach is to compute the fourth power of each expression directly. But let us try a slightly different idea. First let us denote the expressions inside the two parentheses by x and y. That is let x be the quantity square<unk> of 5 +<unk> 3 -<unk> 2 /<unk> 2 and let y be the quantity square<unk> of 5 -<unk> 3 +<unk> 2 /<unk> 2. What we want to find is x 4 + y 4th.

[00:01:10]To compute this, let us first find the values of x + y and x * y. 4x + y. The square<unk> of 3 and the square<unk> of 2 cancel out. The numerator becomes 2 * the square<unk> of 5 and the denominator is square<unk> of 2. After rationalizing the denominator, we find that the value of this expression is square<unk> of 10.

[00:01:37]Next, let us compute x * y. The numerator has the form of a product of a sum and a difference. So we only need to subtract the square of the quantity square<unk> of 3 -<unk> of 2 from the square of square<unk> of 5. Doing so leaves only 2 * the square<unk> of 6 in the numerator. Since the denominator is 2, the value of this expression is square<unk> of 6. Now we have obtained the values of the elementary symmetric expressions x + y and x * y. Rather than computing x 4 + y 4th immediately, it is easier to first find x^2 + y^ 2. Since x^2 + y^ 2 can be rewritten as the square of x + y - 2 * x * y, substituting the values found above gives 10 - 2 *<unk> 6. We use a similar transformation for x 4 + y 4th. It can be rewritten as the square of x^ 2 + y^ 2 - 2 * the square of x * y.

[00:02:50]Substituting the values obtained above, we get the square of the quantity 10 - 2 * the square<unk> of 6 - 2 * the square of the square<unk> 6. Carrying out the calculation, we obtain 112 - 40 * the<unk> 6. That concludes today's lesson.