Solved on Feb 27, 2024

Solve the equation $\sqrt{x^{3}-2}=5$ exactly, using an inverse function when appropriate.

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. We will use inverse operations to isolate $x$ .
3. The square root function has an inverse function, which is squaring both sides of the equation.

STEP 2

To remove the square root, we need to square both sides of the equation.
$(\sqrt{x^{3} - 2})^2 = 5^2$

STEP 3

Squaring the left side removes the square root, and squaring the right side gives us 25.
$x^{3} - 2 = 25$

STEP 4

Now, we need to isolate $x^3$ by moving the constant term to the other side of the equation.
$x^{3} - 2 + 2 = 25 + 2$

STEP 5

Simplify both sides of the equation.
$x^{3} = 27$

STEP 6

To solve for $x$ , we need to take the cube root of both sides of the equation.
$\sqrt[3]{x^{3}} = \sqrt[3]{27}$

STEP 7

Taking the cube root of $x^3$ gives us $x$ , and the cube root of 27 is 3.
$x = 3$
The solution to the equation is $x = 3$ .

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Solve the equation $\sqrt{x^{3}-2}=5$ exactly, using an inverse function when appropriate.

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. We will use inverse operations to isolate $x$ .
3. The square root function has an inverse function, which is squaring both sides of the equation.

STEP 2

To remove the square root, we need to square both sides of the equation.
$(\sqrt{x^{3} - 2})^2 = 5^2$

STEP 3

Squaring the left side removes the square root, and squaring the right side gives us 25.
$x^{3} - 2 = 25$

STEP 4

Now, we need to isolate $x^3$ by moving the constant term to the other side of the equation.
$x^{3} - 2 + 2 = 25 + 2$

STEP 5

Simplify both sides of the equation.
$x^{3} = 27$

STEP 6

To solve for $x$ , we need to take the cube root of both sides of the equation.
$\sqrt[3]{x^{3}} = \sqrt[3]{27}$

STEP 7

Taking the cube root of $x^3$ gives us $x$ , and the cube root of 27 is 3.
$x = 3$
The solution to the equation is $x = 3$ .

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Solve the equation x3−2=5\sqrt{x^{3}-2}=5x3−2​=5 exactly, using an inverse function when appropriate.

STEP 1

Assumptions1. We are solving the equation for xxx.2. We will use inverse operations to isolate xxx.3. The square root function has an inverse function, which is squaring both sides of the equation.

STEP 2

To remove the square root, we need to square both sides of the equation.(x3−2)2=52 (\sqrt{x^{3} - 2})^2 = 5^2 (x3−2​)2=52

STEP 3

Squaring the left side removes the square root, and squaring the right side gives us 25.x3−2=25 x^{3} - 2 = 25 x3−2=25

STEP 4

Now, we need to isolate x3x^3x3 by moving the constant term to the other side of the equation.x3−2+2=25+2 x^{3} - 2 + 2 = 25 + 2 x3−2+2=25+2

STEP 5

Simplify both sides of the equation.x3=27 x^{3} = 27 x3=27

STEP 6

To solve for xxx, we need to take the cube root of both sides of the equation.x33=273 \sqrt[3]{x^{3}} = \sqrt[3]{27} 3x3​=327​

STEP 7

Taking the cube root of x3x^3x3 gives us xxx, and the cube root of 27 is 3.x=3 x = 3 x=3The solution to the equation is x=3x = 3x=3.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation $\sqrt{x^{3}-2}=5$ exactly, using an inverse function when appropriate.

Assumptions
1. We are solving the equation for $x$ .
2. We will use inverse operations to isolate $x$ .
3. The square root function has an inverse function, which is squaring both sides of the equation.

To remove the square root, we need to square both sides of the equation.
$(\sqrt{x^{3} - 2})^2 = 5^2$

Squaring the left side removes the square root, and squaring the right side gives us 25.
$x^{3} - 2 = 25$

Now, we need to isolate $x^3$ by moving the constant term to the other side of the equation.
$x^{3} - 2 + 2 = 25 + 2$

Simplify both sides of the equation.
$x^{3} = 27$

To solve for $x$ , we need to take the cube root of both sides of the equation.
$\sqrt[3]{x^{3}} = \sqrt[3]{27}$

Taking the cube root of $x^3$ gives us $x$ , and the cube root of 27 is 3.
$x = 3$
The solution to the equation is $x = 3$ .