Solved on Jan 29, 2024

Solve the equation (x1)2/3=9(x-1)^{2/3} = 9. Select the correct solution: A. The solution set is {x}\{x\}, where x=93+1x = \sqrt[3]{9} + 1. B. The solution set is the empty set.

STEP 1

Assumptions
1. We are given the equation (x1)2/3=9(x-1)^{2/3} = 9.
2. We need to solve for xx.
3. We assume xx is a real number.

STEP 2

To solve the equation, we start by isolating the term with the variable xx. In this case, the term (x1)2/3(x-1)^{2/3} is already isolated.

STEP 3

To remove the fractional exponent, we can raise both sides of the equation to the reciprocal of the fraction, which is 3/23/2.
(x1)(2/3)(3/2)=93/2(x-1)^{(2/3) \cdot (3/2)} = 9^{3/2}

STEP 4

Simplify the left side of the equation by using the property of exponents that states (am)n=amn(a^{m})^{n} = a^{m \cdot n}.
(x1)1=93/2(x-1)^{1} = 9^{3/2}

STEP 5

The left side simplifies to x1x-1 since any number raised to the power of 1 is itself.
x1=93/2x-1 = 9^{3/2}

STEP 6

Calculate the right side of the equation. The exponent 3/23/2 means we need to take the square root of 9 and then cube it.
93/2=(9)39^{3/2} = (\sqrt{9})^3

STEP 7

Calculate the square root of 9.
9=3\sqrt{9} = 3

STEP 8

Now cube the result from STEP_7.
33=273^3 = 27

STEP 9

Substitute the result from STEP_8 into the equation.
x1=27x-1 = 27

STEP 10

Add 1 to both sides of the equation to solve for xx.
x=27+1x = 27 + 1

STEP 11

Calculate the value of xx.
x=28x = 28

STEP 12

Now we need to check if our solution is valid by substituting it back into the original equation.
(281)2/3=9(28-1)^{2/3} = 9

STEP 13

Simplify the left side of the equation.
272/3=927^{2/3} = 9

STEP 14

Since 27=3327 = 3^3, we can rewrite the left side using the property of exponents.
(33)2/3=9(3^3)^{2/3} = 9

STEP 15

Simplify the left side by multiplying the exponents.
332/3=93^{3 \cdot 2/3} = 9

STEP 16

Simplify the exponent on the left side.
32=93^2 = 9

STEP 17

Calculate the left side.
9=99 = 9

STEP 18

Since both sides of the equation are equal, our solution x=28x = 28 is valid.
The solution set is {28}\{28\}.
A. The solution set is {28}\{28\}.

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