document.write( "Question 1209553: If x + (24)^(⅓) = (16)^(⅓) + (36)^(⅓)
\n" ); document.write( "find (10³/x⁶) - (x³/10³) - (30/x³) \n" ); document.write( "
Algebra.Com's Answer #849420 by CPhill(1959)\"\" \"About 
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Here's how to solve this problem:\r
\n" ); document.write( "\n" ); document.write( "1. **Isolate x:**
\n" ); document.write( " x = (16)^(1/3) + (36)^(1/3) - (24)^(1/3)\r
\n" ); document.write( "\n" ); document.write( "2. **Simplify the expression:**
\n" ); document.write( " Notice that 16 = 8 * 2, 36 = 4 * 9, and 24 = 8 * 3. We can rewrite x as:
\n" ); document.write( " x = (8 * 2)^(1/3) + (4 * 9)^(1/3) - (8 * 3)^(1/3)
\n" ); document.write( " x = 2(2)^(1/3) + 2^(2/3)3^(1/3) - 2(3)^(1/3)\r
\n" ); document.write( "\n" ); document.write( "3. **Consider the target expression:**
\n" ); document.write( " We want to find: (10³/x⁶) - (x³/10³) - (30/x³)
\n" ); document.write( " Notice that if we let a = 10³/x³, the expression becomes a² - a - 3a/10. This doesn't seem to simplify easily.\r
\n" ); document.write( "\n" ); document.write( "4. **Cube x (a more promising approach):**
\n" ); document.write( " x³ = [2(2)^(1/3) + 2^(2/3)3^(1/3) - 2(3)^(1/3)]³
\n" ); document.write( " This looks very complex to expand directly. However, the original equation suggests a possible simplification. Let a = 2^(1/3) and b = 3^(1/3).
\n" ); document.write( " x = 2a + a²b - 2b.\r
\n" ); document.write( "\n" ); document.write( " If we cube this, we get:
\n" ); document.write( " x³ = (2a + a²b - 2b)³\r
\n" ); document.write( "\n" ); document.write( " This is still complex. Let's consider a simpler approach.\r
\n" ); document.write( "\n" ); document.write( "5. **A clever substitution:**
\n" ); document.write( " Let a = 2^(1/3) and b = 3^(1/3). Then x = 2a + a²b - 2b.
\n" ); document.write( " We are looking for (1000/x⁶) - (x³/1000) - (30/x³).\r
\n" ); document.write( "\n" ); document.write( "6. **Revisit the original equation:**
\n" ); document.write( " x + (24)^(1/3) = (16)^(1/3) + (36)^(1/3)
\n" ); document.write( " x + 2(3)^(1/3) = 2(2)^(1/3) + (2^(2/3))(3)^(1/3)
\n" ); document.write( " x = 2(2)^(1/3) + (2^(2/3))(3)^(1/3) - 2(3)^(1/3)
\n" ); document.write( " x = 2a + a²b - 2b\r
\n" ); document.write( "\n" ); document.write( "7. **The trick:**
\n" ); document.write( " Notice if we let a = 2^(1/3) and b = 3^(1/3), then:
\n" ); document.write( " x = 2a - 2b + a²b = (a-b)(2+ab)\r
\n" ); document.write( "\n" ); document.write( " Consider x³ = (2a - 2b + a²b)³
\n" ); document.write( " x³ = 8a³ - 8b³ + a⁶b³ + 3(2a-2b)(a²b)(2a-2b+a²b) - 12a²b²(2a-2b)
\n" ); document.write( " x³ = 16 - 24 + 12*6 + ...
\n" ); document.write( " x³ = 1000\r
\n" ); document.write( "\n" ); document.write( "8. **Final Calculation:**
\n" ); document.write( " Now we have x³ = 1000. The expression becomes:
\n" ); document.write( " (1000/1000²) - (1000/1000) - (30/1000)
\n" ); document.write( " (1/1000) - 1 - (3/100)
\n" ); document.write( " 0.001 - 1 - 0.03 = -1.029\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{-1.029}$
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