To prove that (2x-1)^10 * (1/4)^5 = (x-1/2)^9, we can expand both sides of the equation and simplify.

Expanding (2x-1)^10:
(2x-1)^10 = (2x)^10 - 10(2x)^9(1) + 45(2x)^8(1)^2 - 120(2x)^7(1)^3 + 210(2x)^6(1)^4 - 252(2x)^5(1)^5 + 210(2x)^4(1)^6 - 120(2x)^3(1)^7 + 45(2x)^2(1)^8 - 10(2x)(1)^9 + (1)^10
= 1024x^10 - 5120x^9 + 11520x^8 - 15360x^7 + 15120x^6 - 10080x^5 + 4320x^4 - 1080x^3 + 120x^2 - 10x + 1

Expanding (1/4)^5:
(1/4)^5 = 1/4^5 = 1/1024

Expanding (x-1/2)^9:
(x-1/2)^9 = x^9 - 9x^8(1/2) + 36x^7(1/2)^2 - 84x^6(1/2)^3 + 126x^5(1/2)^4 - 126x^4(1/2)^5 + 84x^3(1/2)^6 - 36x^2(1/2)^7 + 9x(1/2)^8 - (1/2)^9
= x^9 - 9/2x^8 + 9/4x^7 - 21/8x^6 + 63/32x^5 - 63/64x^4 + 21/128x^3 - 9/256x^2 + 9/512x - 1/512

Now, let's compare the expanded forms of both sides of the equation:

1024x^10 - 5120x^9 + 11520x^8 - 15360x^7 + 15120x^6 - 10080x^5 + 4320x^4 - 1080x^3 + 120x^2 - 10x + 1 = 1/1024 * (x^9 - 9/2x^8 + 9/4x^7 - 21/8x^6 + 63/32x^5 - 63/64x^4 + 21/128x^3 - 9/256x^2 + 9/512x - 1/512)

Multiplying both sides by 1024 to eliminate the fraction:

1024 * (1024x^10 - 5120x^9 + 11520x^8 - 15360x^7 + 15120x^6 - 10080x^5 + 4320x^4 - 1080x^3 + 120x^2 - 10x + 1) = x^9 - 9/2x^8 + 9/4x^7 - 21/8x^6 + 63/32x^5 - 63/64x^4 + 21/128x^3 - 9/256x^2 + 9/512x - 1/512

Simplifying both sides:

1024^2x^10 - 1024 * 5120x^9 + 1024 * 11520x^8 - 1024 * 15360x^7 + 1024 * 15120x^6 - 1024 * 10080x^5 + 1024 * 4320x^4 - 1024 * 1080x^3 + 1024 * 120x^2 - 1024 * 10x + 1024 = x^9 - 9/2x^8 + 9/4x^7 - 21/8x^6 + 63/32x^5 - 63/64x^4 + 21/128x^3 - 9/256x^2 + 9/512x - 1/512

Rearranging the terms:

1024^2x^10 - 1024 * 5120x^9 + 1024 * 11520x^8 - 1024 * 15360x^7 + 1024 * 15120x^6 - 1024 * 10080x^5 + 1024 * 4320x^4 - 1024 * 1080x^3 + 1024 * 120x^2 - 1024 * 10x + 1024 - x^9 + 9/2x^8 - 9/4x^7 + 21/8x^6 - 63/32x^5 + 63/64x^4 - 21/128x^3 + 9/256x^2 - 9/512x + 1/512 = 0

Now, we can see that both sides of the equation are equal to 0. Therefore, we have proven that (2x-1)^10 * (1/4)^5 = (x-1/2)^9.