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Answer :
To solve the equation [tex]\((3x + 4)^3 = 2197\)[/tex], we can take the following steps:
1. Understand the Equation: The equation given is [tex]\((3x + 4)^3 = 2197\)[/tex]. We need to find the value(s) of [tex]\(x\)[/tex] that satisfy this equation.
2. Recognize the Simplification: Notice that 2197 is a perfect cube. Specifically, [tex]\(13^3 = 2197\)[/tex].
3. Equal the Inside Expression: Since [tex]\(13^3 = 2197\)[/tex], we can set:
[tex]\[
3x + 4 = 13
\][/tex]
This is because if [tex]\((3x + 4)^3 = 13^3\)[/tex], then the expressions inside the parentheses must be equal.
4. Solve for [tex]\(x\)[/tex]:
- From [tex]\(3x + 4 = 13\)[/tex], subtract 4 from both sides:
[tex]\[
3x = 9
\][/tex]
- Next, divide both sides by 3 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 3
\][/tex]
5. Consider Complex Solutions: The cubic equation could potentially have complex solutions, but in this context, the real-value solution for [tex]\(x\)[/tex] is primary. For those wondering about the nature of complex roots, they result from the symmetric property of cubics around the real roots. However, common practice for such problems is focusing on the real root which, in this case, is [tex]\(x = 3\)[/tex].
So, the solution to the equation [tex]\((3x + 4)^3 = 2197\)[/tex] is [tex]\(x = 3\)[/tex].
1. Understand the Equation: The equation given is [tex]\((3x + 4)^3 = 2197\)[/tex]. We need to find the value(s) of [tex]\(x\)[/tex] that satisfy this equation.
2. Recognize the Simplification: Notice that 2197 is a perfect cube. Specifically, [tex]\(13^3 = 2197\)[/tex].
3. Equal the Inside Expression: Since [tex]\(13^3 = 2197\)[/tex], we can set:
[tex]\[
3x + 4 = 13
\][/tex]
This is because if [tex]\((3x + 4)^3 = 13^3\)[/tex], then the expressions inside the parentheses must be equal.
4. Solve for [tex]\(x\)[/tex]:
- From [tex]\(3x + 4 = 13\)[/tex], subtract 4 from both sides:
[tex]\[
3x = 9
\][/tex]
- Next, divide both sides by 3 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 3
\][/tex]
5. Consider Complex Solutions: The cubic equation could potentially have complex solutions, but in this context, the real-value solution for [tex]\(x\)[/tex] is primary. For those wondering about the nature of complex roots, they result from the symmetric property of cubics around the real roots. However, common practice for such problems is focusing on the real root which, in this case, is [tex]\(x = 3\)[/tex].
So, the solution to the equation [tex]\((3x + 4)^3 = 2197\)[/tex] is [tex]\(x = 3\)[/tex].
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