7^x+2-6cdot 7^x+1+3cdot 7^x=10

To solve the equation \(7^{x+2} - 6 \cdot 7^{x+1} + 3 \cdot 7^x = 10\), let's start by simplifying the expression using properties of exponents.<br /><br />First, rewrite each term with a common base:<br />\[7^{x+2} = 7^2 \cdot 7^x = 49 \cdot 7^x\]<br />\[6 \cdot 7^{x+1} = 6 \cdot 7 \cdot 7^x = 42 \cdot 7^x\]<br />\[3 \cdot 7^x = 3 \cdot 7^x\]<br /><br />Now substitute these into the original equation:<br />\[49 \cdot 7^x - 42 \cdot 7^x + 3 \cdot 7^x = 10\]<br /><br />Combine like terms:<br />\[(49 - 42 + 3) \cdot 7^x = 10\]<br />\[10 \cdot 7^x = 10\]<br /><br />Divide both sides by 10:<br />\[7^x = 1\]<br /><br />Since \(7^0 = 1\), we have:<br />\[x = 0\]<br /><br />Thus, the solution to the equation is:<br />\[x = 0\]