Simplify: Nested Powers

What you do

You shape the function y = (k*x^5)^2 / x^m and watch the graph answer.

Challenges to clear

• Set k = 2 and slide m to 10: at x = 2, (2·32)² / 2¹⁰ = 4096/1024 = 4.
• The curve goes FLAT at 4: x = 3 gives 4 too — every x cancelled.
• A fourth power inside the bracket now. Set the coefficient k to 3.
• Squaring gives x⁸ on top. Slide m until the curve goes FLAT at 9 — every x has cancelled, leaving just k².

Check yourself

Why does the expression (2 * x^5)^2 / x^10 always equal 4, regardless of the value of x?

In the step (2 * x^5)^2, why does the coefficient 2 become 4?

• Because (2*x^5)^2 becomes 4*x^10, and x^10 divided by x^10 is 1, leaving only 4. — correct
• Because x^10 in the denominator cancels out the 2 in the numerator.
• Because squaring x^5 makes it x^10, which is larger than the denominator, so it grows.
• Because 2 squared is 4, and the x terms are ignored in division.
• The exponent applies to every factor inside the parentheses, so 2^2 = 4. — correct
• The exponent only applies to the variable x, not the number 2.
• It is a special rule that coefficients double when squared.
• The 2 is added to the exponent 5, resulting in 7, then squared.

Think about it

• (2x⁵)² = 4x¹⁰. Dividing by x^m — what m kills ALL the x's?