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Sure, let's solve the equation [tex]\( 6 + \sqrt[5]{249 - 2x} = 7 \)[/tex] step by step.
1. Isolate the radical: We start by isolating the fifth root term on one side of the equation. To do this, we can subtract 6 from both sides of the equation: [tex]\[
6 + \sqrt[5]{249 - 2x} - 6 = 7 - 6
\][/tex] Simplifying this: [tex]\[
\sqrt[5]{249 - 2x} = 1
\][/tex]
2. Eliminate the radical: Next, we need to get rid of the fifth root by raising both sides of the equation to the power of 5: [tex]\[
\left(\sqrt[5]{249 - 2x}\right)^5 = 1^5
\][/tex] This simplifies to: [tex]\[
249 - 2x = 1
\][/tex]
3. Solve for [tex]\( x \)[/tex]: Now, we solve for [tex]\( x \)[/tex]. Start by isolating the term that contains [tex]\( x \)[/tex]: [tex]\[
249 - 2x = 1
\][/tex] Subtract 249 from both sides: [tex]\[
-2x = 1 - 249
\][/tex] Simplify the right-hand side: [tex]\[
-2x = -248
\][/tex] Finally, divide both sides by -2: [tex]\[
x = \frac{-248}{-2}
\][/tex] Simplifying this gives: [tex]\[
x = 124
\][/tex]
Thus, the solution to the equation [tex]\( 6 + \sqrt[5]{249 - 2x} = 7 \)[/tex] is [tex]\( x = 124 \)[/tex].
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