Problem 51
Question
To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \(\left.(\sqrt[n]{y})^{n}=y .\right)\). $$f(x)=\sqrt[3]{x}+1$$
Step-by-Step Solution
Verified Answer
The required solution is \(x = f(x)^3 - 3f(x)^2 + 3f(x) - 1\).
1Step 1: Isolate the nth root
Start by subtracting 1 from both sides of the equation \(f(x) = \sqrt[3]{x} + 1\), which gives us \(f(x) - 1 = \sqrt[3]{x}\).
2Step 2: Remove the cube root
Using the rule \(\left.(\sqrt[n]{y})^{n}=y .\right)\) where \(n = 3\) and \(y = f(x) - 1\), we can remove the cube root by cubing both sides of the equation: \((f(x) - 1)^3 = x\).
3Step 3: Expand the Cube
From previous step we have \( (f(x) - 1)^3 = x \) . Now expand the cube \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Thus \((f(x) - 1)^3\) becomes \(x = f(x)^3 - 3f(x)^2 + 3f(x) - 1\) .
Key Concepts
Understanding nth RootsIsolating VariablesExpanding Binomials
Understanding nth Roots
The concept of an nth root is essential in mathematics, especially when dealing with roots and powers. The nth root of a number y is the number x that, when raised to the power n, yields y. Mathematically, this is expressed as \(x = \sqrt[n]{y}\). Here the 'n' is what determines which root you are dealing with, such as square root \(\sqrt{y}\) for n=2, or cube root \(\sqrt[3]{y}\) for n=3.
Nth roots are useful for solving equations where the variable is under a root, such as \(f(x)=\sqrt[3]{x}+1\). In problems like these, our goal is to isolate the term with the variable inside the root. Once isolated, we can remove the root by raising both sides of the equation to the nth power, efficiently canceling out the root. This step transforms the nth root into a simple linear or polynomial expression, which is much easier to solve.
Nth roots are useful for solving equations where the variable is under a root, such as \(f(x)=\sqrt[3]{x}+1\). In problems like these, our goal is to isolate the term with the variable inside the root. Once isolated, we can remove the root by raising both sides of the equation to the nth power, efficiently canceling out the root. This step transforms the nth root into a simple linear or polynomial expression, which is much easier to solve.
Isolating Variables
Isolating variables is a crucial step in solving mathematical equations. It involves rearranging the equation so that the variable we are solving for is on one side, alone. This makes it easier to see what each step of the solution involves.
In our example, the function is \(f(x) = \sqrt[3]{x} + 1\). To solve for \(x\), we start by getting rid of the constant term on the same side as the root. Subtracting 1 from both sides gives us \(f(x) - 1 = \sqrt[3]{x}\). This isolates the term \(\sqrt[3]{x}\), enabling us to focus on removing the cube root next. Steps like these are important because they simplify equations, making them easier to handle.
In our example, the function is \(f(x) = \sqrt[3]{x} + 1\). To solve for \(x\), we start by getting rid of the constant term on the same side as the root. Subtracting 1 from both sides gives us \(f(x) - 1 = \sqrt[3]{x}\). This isolates the term \(\sqrt[3]{x}\), enabling us to focus on removing the cube root next. Steps like these are important because they simplify equations, making them easier to handle.
Expanding Binomials
Once the root has been removed from an equation, expanding binomials becomes a necessary skill. This expansion involves turning expressions like \((a - b)^3\) into a polynomial form. The expansion formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) is used to break down the binomial into terms that are easier to work with.
In the step where we have \((f(x) - 1)^3 = x\), expanding the cube uses the known formula to result in \(f(x)^3 - 3f(x)^2 + 3f(x) - 1 = x\). This polynomial gives us a clearer view of the equation and is easier to differentiate, integrate, or graph if needed. Expanding binomials helps convert a complex equation into a series of simpler terms, facilitating better understanding and manipulation of mathematical problems.
In the step where we have \((f(x) - 1)^3 = x\), expanding the cube uses the known formula to result in \(f(x)^3 - 3f(x)^2 + 3f(x) - 1 = x\). This polynomial gives us a clearer view of the equation and is easier to differentiate, integrate, or graph if needed. Expanding binomials helps convert a complex equation into a series of simpler terms, facilitating better understanding and manipulation of mathematical problems.
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