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Chapter 2: Problem 43

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[3]{x+3}=5$$

Short Answer

Expert verified
The real solution for the given radical equation is \(x=122\).

Step by step solution

01

Isolate the cube root

The first step to isolate 'x' is to remove the cube root. To do this, we can raise each side of the equation to the power of three. This gives us the following equation: \((\sqrt[3]{x+3})^3=5^3\).
02

Calculate the cube of 5

Next, calculate the cube of 5. This calculation is \(5^3 = 125\). So the equation becomes: \(x+3 = 125\).
03

Solve for 'x'

To solve the equation 'x + 3 = 125' for 'x', subtract 3 from both sides of the equation. This gives us the solution: \(x = 125 - 3 = 122\).
04

Check the solution

Finally, check the solution by substituting 'x' into the original equation. This gives us \(\sqrt[3]{122+3}\) which simplifies to \(\sqrt[3]{125}\), and that is equal to 5. Since the left-hand side equals the right-hand side, your solution checks out.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Root Isolation
When solving radical equations involving a cube root, it's necessary to first isolate the cube root expression on one side of the equation.

Isolating the cube root makes it possible to then eliminate the cube root by raising both sides of the equation to the power of three. In our example, the cube root of (\(x+3\)) is isolated and subsequently cubed, which essentially reverses the cube root operation. This step is crucial as it simplifies the equation to a basic algebraic form that is easier to solve.

It is equivalent to 'undoing' the cube root to retrieve the original value that was cubed, to begin with. Always ensure that no other terms or operations are interfering with the cube root before you proceed to the next step.
Exponentiation
Exponentiation is the process of raising a number to a power. In the context of radical equations, when the radical is a cube root, we raise both sides of the equation to the power of three.

By exponentiating the cube root, we effectively remove the radical, allowing us to work with a simpler algebraic equation. In our exercise example, raising the cube root to the third power and the number 5 to the third power leads us to the intermediate equation of \(x + 3 = 125\).

It's essential to perform the same operation on both sides of the equation to maintain equality, which is one of the fundamental principles of algebra.
Real Solutions
Real solutions are the possible values that satisfy an equation in the real number system. When dealing with cube roots, we typically encounter real solutions because cubing a real number will return a real number.

However, it's important to note that not every radical equation will have a real solution. For instance, solving an equation involving even roots (like square roots) could lead to imaginary numbers if the radicand is negative.

In our exercise, after removing the cube root and simplifying the algebra, we find that \(x = 122\), which is a real number, thus, it is a real solution to our equation.
Equation Checking
After solving a radical equation, it is essential to verify that the solution is correct. This process is known as equation checking. It involves substituting the obtained solution back into the original equation to see if it produces a true statement.

For our cube root equation, checking entails substituting \(x\) with 122 and confirming that the cube root of \(122 + 3\) is indeed equal to 5. If both sides of the original equation match after the substitution, then the solution is verified as correct.

This step is vital because it helps to confirm that no extraneous solutions have been included. Extraneous solutions may arise when raising both sides of an equation to an even power, which is not a concern in this cube root scenario.

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Most popular questions from this chapter

The shape of the Gateway Arch in St. Louis, Missouri, can be approximated by a parabola. (The actual shape will be discussed in an exercise in the chapter on exponential functions.) The highest point of the arch is approximately 625 feet above the ground, and the arch has a (horizontal) span of approximately 600 feet. (Check your book to see image) (a) Set up a coordinate system with the origin at the midpoint of the base of the arch. What are the \(x\) -intercepts of the parabola, and what do they represent? (b) What do the \(x\)- and \(y\)-coordinates of the vertex of the parabola represent? (c) Write an expression for the quadratic function associated with the parabola. (Hint: Use the vertex form of a quadratic function and find the coefficient \(a .)\) (d) What is the \(y\)-coordinate of a point on the arch whose (horizontal) distance from the axis of symmetry of the parabola is 100 feet?

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((f \circ f)(t)\), and give the domain of \(f \circ f\)

A parabola associated with a certain quadratic function \(f\) has the point (2,8) as its vertex and passes through the point \((4,0) .\) Find an expression for \(f(x)\) in the form \(f(x)=a(x-h)^{2}+k\) (a) From the given information, find the values of \(h\) and \(k\) (b) Substitute the values you found for \(h\) and \(k\) into the expression \(f(x)=a(x-h)^{2}+k\) (c) Now find \(a\). To do this, use the fact that the parabola passes through the point \((4,0) .\) That is, \(f(4)=0\) You should get an equation having just \(a\) as a variable. Solve for \(a\) (d) Substitute the value you found for \(a\) into the expression you found in part (b). (e) Graph the function using a graphing utility and check your answer. Is (2,8) the vertex of the parabola? Does the parabola pass through (4,0)\(?\)

The following table gives the average hotel room rate for selected years from 1990 to \(1999 .\) (Source:American Hotel and Motel Association) $$\begin{array}{cc}\text { Year } & \text { Rate (in dollars) } \\\\\hline 1990 & 57.96 \\\1992 & 58.91 \\\1994 & 62.86 \\\1996 & 70.93 \\\1998 & 78.62 \\\1999 & 81.33\end{array}$$ (a) What general trend do you notice in these figures? (b) Fit both a linear and a quadratic function to this set of points, using the number of years since 1990 as the independent variable. (c) Based on your answer to part (b), which function would you use to model this set of data, and why? (d) Using the quadratic model, find the year in which the average hotel room rate will be \(\$ 85\)

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions of dollars. The model is based on data for the years \(1981-2000 .\) When did the attendance reach \(\$ 12\) million? (Source: The League of American Theaters and Producers, Inc.)
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