Problem 38 - College Algebra Essentials

Question

Solve each equation with rational exponents. Check all proposed solutions. $$ (x+5)^{\frac{2}{3}}=4 $$

Step-by-Step Solution

Answer

The solutions to the given equation are \( x=59 \) and \( x=-69 \).

Step 1: Remove the cube root

By raising both sides of the equation to the power of \(\frac{3}{2}\), we have \( (x+5)=4^{\frac{3}{2}} \)

Step 2: Square the equation

Subsequently, square both sides of the equation to get \( (x+5)^2=4^3 \)

Step 3: Simplify and solve the equation

Now, simplify and solve the equation, which is \( x+5=\sqrt[3]{64} \) or \( x+5=-\sqrt[3]{64} \). Solve to get \( x=\sqrt[3]{64}-5 \) and \( x=-\sqrt[3]{64}-5 \). The solutions are \( x=59 \) and \( x=-69 \).

Step 4: Check the solutions

Substitute the two obtained solutions \( x=59 \) and \( x=-69 \) back into the original equation to verify whether they satisfy the original equation or not. Through substitution, we can confirm that indeed \( (59+5)^{\frac{2}{3}} =4 \) and \( (-69+5)^{\frac{2}{3}} =4 \). Hence, the solutions \( x=59 \) and \( x=-69 \) are valid.

Key Concepts

Equation Solving

When solving equations involving rational exponents, it's essential to understand that these expressions can be rewritten in a more accessible form. Rational exponents are another way of describing roots and powers in one term. For example, an expression like \((x+5)^{\frac{2}{3}}\) means we first take the cube root of \(x+5\) and then square the result.

Here are key steps in solving these kinds of equations:

• First, isolate the term with the rational exponent if needed.
• Raise both sides of the equation to a power that eliminates the fraction in the exponent. This could involve multiplication by the reciprocal of the rational exponent.
• Solve the resulting simpler equation by traditional algebraic methods.

In our example, after isolating the expression \((x+5)^{\frac{2}{3}}\), raising both sides to the \(\frac{3}{2}\) power helps simplify the problem, leading to the equation \((x+5)^2 = 4^3\), which is easier to tackle.

Cube Roots

The cube root is the operation that answers the question: What number when cubed gives the original number? In symbols, the cube root of a number \(a\) is \(\sqrt[3]{a}\). This is a crucial concept when dealing with equations involving cubes or cube roots.

To solve our problem, after simplifying the equation to \((x + 5)^2 = 4^3\), you find expressions of the form \(x + 5 = \sqrt[3]{64}\) or \(x + 5 = -\sqrt[3]{64}\), realizing that 4 cubed is 64 because \(4^3 = 64\).

• Cube roots can have both positive and negative results since both \((\sqrt[3]{64})\) and \((-\sqrt[3]{64})\) are valid approaches to reaching a cube value like 64 because \((-4)^3\) and \(4^3\) both equal 64.
• It is significant to consider all possibilities with cube roots. Hence, when solving equations, remember to include negative cube roots where relevant.

Verification of Solutions

Checking solutions is crucial to ensure the correctness of the values obtained from solving equations. Several factors, like mistaken calculations or misunderstood steps, can affect the solution's accuracy.

To perform verification:

• Start by substituting the solution back into the original equation.
• Ensure the left-hand side equals the right-hand side of the equation.
• Verify all possible solutions, because rational exponents can sometimes lead to multiple valid answers.

In our example, the solutions \(x = 59\) and \(x = -69\) need checking by plugging them back into the equation \((x+5)^{\frac{2}{3}} = 4\). By calculating:

• When \(x = 59\), \((59 + 5)^{\frac{2}{3}}\) simplifies to \(4\).
• When \(x = -69\), \((-69 + 5)^{\frac{2}{3}}\) also simplifies to \(4\).

Both solutions satisfy the original equation, confirming their validity.