Answer :
Let's solve these equations step-by-step.
### Solving Equation (5): [tex]\(2 x^3 = 54\)[/tex]
1. Isolate the [tex]\(x^3\)[/tex] term:
[tex]\[ 2x^3 = 54 \][/tex]
Divide both sides by 2:
[tex]\[ x^3 = \frac{54}{2} \][/tex]
[tex]\[ x^3 = 27 \][/tex]
2. Find the cube root of both sides:
[tex]\[ x = \sqrt[3]{27} \][/tex]
The cube roots of 27 are the solutions to the equation, and they include real and complex roots. The primary real solution is:
[tex]\[ x = 3 \][/tex]
Considering the properties of cube roots for complex numbers, the other two roots are:
[tex]\[ x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2} \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
So, the solutions to [tex]\(2 x^3 = 54\)[/tex] are:
[tex]\[ x = 3, \quad x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
### Solving Equation (7): [tex]\(n^{-\frac{2}{3}} = 9\)[/tex]
1. Rewrite the equation using positive exponents:
[tex]\[ n^{-\frac{2}{3}} = 9 \][/tex]
Rewriting the negative exponent:
[tex]\[ n^{-\frac{2}{3}} = \frac{1}{n^{\frac{2}{3}}} \][/tex]
Thus:
[tex]\[ \frac{1}{n^{\frac{2}{3}}} = 9 \][/tex]
By taking reciprocals of both sides:
[tex]\[ n^{\frac{2}{3}} = \frac{1}{9} \][/tex]
2. Solving for [tex]\(n\)[/tex]:
Raise both sides to the power of [tex]\(\frac{3}{2}\)[/tex] to solve for [tex]\(n\)[/tex]:
[tex]\[ \left(n^{\frac{2}{3}}\right)^{\frac{3}{2}} = \left(\frac{1}{9}\right)^{\frac{3}{2}} \][/tex]
This simplifies to:
[tex]\[ n = \left(\frac{1}{9}\right)^{\frac{3}{2}} \][/tex]
Evaluate [tex]\(\left(\frac{1}{9}\right)^{\frac{3}{2}}\)[/tex]:
[tex]\[ \left(\frac{1}{9}\right)^{\frac{3}{2}} = \left(\frac{1}{3^2}\right)^{\frac{3}{2}} = \left(\frac{1}{3^2}\right)^{\frac{3}{2}} = \frac{1}{3^3} = \frac{1}{27} \][/tex]
So, the solution to [tex]\(n^{-\frac{2}{3}} = 9\)[/tex] is:
[tex]\[ n = \frac{1}{27} = 0.0370370370370370 \][/tex]
### Summary of Solutions
For the equation [tex]\(2 x^3 = 54\)[/tex]:
[tex]\[ x = 3, \quad x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
For the equation [tex]\(n^{-\frac{2}{3}} = 9\)[/tex]:
[tex]\[ n = 0.0370370370370370 (\frac{1}{27}) \][/tex]
### Solving Equation (5): [tex]\(2 x^3 = 54\)[/tex]
1. Isolate the [tex]\(x^3\)[/tex] term:
[tex]\[ 2x^3 = 54 \][/tex]
Divide both sides by 2:
[tex]\[ x^3 = \frac{54}{2} \][/tex]
[tex]\[ x^3 = 27 \][/tex]
2. Find the cube root of both sides:
[tex]\[ x = \sqrt[3]{27} \][/tex]
The cube roots of 27 are the solutions to the equation, and they include real and complex roots. The primary real solution is:
[tex]\[ x = 3 \][/tex]
Considering the properties of cube roots for complex numbers, the other two roots are:
[tex]\[ x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2} \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
So, the solutions to [tex]\(2 x^3 = 54\)[/tex] are:
[tex]\[ x = 3, \quad x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
### Solving Equation (7): [tex]\(n^{-\frac{2}{3}} = 9\)[/tex]
1. Rewrite the equation using positive exponents:
[tex]\[ n^{-\frac{2}{3}} = 9 \][/tex]
Rewriting the negative exponent:
[tex]\[ n^{-\frac{2}{3}} = \frac{1}{n^{\frac{2}{3}}} \][/tex]
Thus:
[tex]\[ \frac{1}{n^{\frac{2}{3}}} = 9 \][/tex]
By taking reciprocals of both sides:
[tex]\[ n^{\frac{2}{3}} = \frac{1}{9} \][/tex]
2. Solving for [tex]\(n\)[/tex]:
Raise both sides to the power of [tex]\(\frac{3}{2}\)[/tex] to solve for [tex]\(n\)[/tex]:
[tex]\[ \left(n^{\frac{2}{3}}\right)^{\frac{3}{2}} = \left(\frac{1}{9}\right)^{\frac{3}{2}} \][/tex]
This simplifies to:
[tex]\[ n = \left(\frac{1}{9}\right)^{\frac{3}{2}} \][/tex]
Evaluate [tex]\(\left(\frac{1}{9}\right)^{\frac{3}{2}}\)[/tex]:
[tex]\[ \left(\frac{1}{9}\right)^{\frac{3}{2}} = \left(\frac{1}{3^2}\right)^{\frac{3}{2}} = \left(\frac{1}{3^2}\right)^{\frac{3}{2}} = \frac{1}{3^3} = \frac{1}{27} \][/tex]
So, the solution to [tex]\(n^{-\frac{2}{3}} = 9\)[/tex] is:
[tex]\[ n = \frac{1}{27} = 0.0370370370370370 \][/tex]
### Summary of Solutions
For the equation [tex]\(2 x^3 = 54\)[/tex]:
[tex]\[ x = 3, \quad x = -\frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad \text{and} \quad x = -\frac{3}{2} + \frac{3\sqrt{3}i}{2} \][/tex]
For the equation [tex]\(n^{-\frac{2}{3}} = 9\)[/tex]:
[tex]\[ n = 0.0370370370370370 (\frac{1}{27}) \][/tex]
