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Answer :
To solve the equation [tex]\frac{6x^6}{4 \cdot x} - 7x^6 = 21[/tex], let's go through the steps to simplify and solve it.
Step 1: Simplify the Expression
The equation is [tex]\frac{6x^6}{4x} - 7x^6 = 21[/tex].
First, we simplify [tex]\frac{6x^6}{4x}[/tex].
- Divide the coefficients: [tex]\frac{6}{4} = \frac{3}{2}[/tex].
- For the exponents, [tex]x^6/x = x^{6-1} = x^5[/tex].
So, [tex]\frac{6x^6}{4x} = \frac{3}{2}x^5[/tex].
Now, the equation becomes:
[tex]\frac{3}{2}x^5 - 7x^6 = 21[/tex]
Step 2: Factor and Rearrange
Rearrange the terms to set the equation to zero:
[tex]\frac{3}{2}x^5 - 7x^6 - 21 = 0[/tex]
Step 3: Finding a Common Factor
There is no straightforward common factor among all terms, so we will try factoring or using different methods such as substitution or numerical methods. For simplicity and checking for elementary solutions, try substitution with simple numbers.
Step 4: Solve for Special Cases
Without further simplification possible by hand, testing specific values could be useful. Note that this is an advanced equation and typical algebraic techniques or numerical solving tools might be required for exact root finding.
For manual solutions, numerical approximations or graphing techniques might suggest potential roots.
Conclusion
Without using a calculator or advanced methods, solving this precisely by hand is challenging given the complexity and powers involved without particular substitutions. Using computers or graphing might show if more apparent roots exist, not easily factored manually with middle school or high school algebra.
In practical scenarios, tools or software might be necessary to find approximate solutions effectively.
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