Problem 39 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \left(m^{3}+26\right)^{1 / 3}=m+2 $$

Step-by-Step Solution

Answer

The solution is \( m = 1 \).

Step 1: Simplify the Equation

Begin by recognizing that the left side of the equation, \( \left(m^3 + 26\right)^{1/3} \), is the cube root of \( m^3 + 26 \). Set this equal to the right side, which is \( m + 2 \). Therefore: \[ \left(m^3 + 26\right)^{1/3} = m + 2 \]

Step 2: Eliminate the Cube Root

To eliminate the cube root, cube both sides of the equation. This gives: \[ m^3 + 26 = (m + 2)^3 \]Next, expand the right side: \[ (m + 2)^3 = m^3 + 6m^2 + 12m + 8 \]

Step 3: Rearrange Terms

Now substitute the expanded form back into the equation: \[ m^3 + 26 = m^3 + 6m^2 + 12m + 8 \]Subtract \( m^3 \) from both sides to simplify:\[ 26 = 6m^2 + 12m + 8 \]

Step 4: Further Simplify

Subtract 8 from both sides to isolate the quadratic terms: \[ 18 = 6m^2 + 12m \]Next, divide the entire equation by 6 to simplify:\[ 3 = m^2 + 2m \]

Step 5: Solve the Quadratic Equation

Rewrite the equation in standard quadratic form:\[ m^2 + 2m - 3 = 0 \]Use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -3 \).

Step 6: Apply the Quadratic Formula

Calculate the discriminant: \[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-3) = 4 + 12 = 16 \]Now find the roots using the quadratic formula:\[ m = \frac{-2 \pm \sqrt{16}}{2 \cdot 1} \]\[ m = \frac{-2 \pm 4}{2} \]

Step 7: Solve for Values of m

Calculate the two potential solutions:1. When \( m = \frac{-2 + 4}{2} = 1 \).2. When \( m = \frac{-2 - 4}{2} = -3 \).

Step 8: Verify Solutions

Substitute each solution back into the original equation to verify:1. For \( m = 1 \): \[ \left(1^3 + 26\right)^{1/3} = 1 + 2 \Rightarrow 3 = 3 \] 2. For \( m = -3 \): \[ ((-3)^3 + 26)^{1/3} = -3 + 2 \Rightarrow 1 = -1 \]Only \( m = 1 \) is a valid solution.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of variables, coefficients, and mathematical operations like addition, subtraction, multiplication, and division. For example, in the expression \( m^3 + 26 \), \( m \) represents the variable, and \( 26 \) is a constant.
When solving equations that include algebraic expressions, it's important to manipulate these expressions to isolate the variable. This means applying operations that will allow you to consolidate or eliminate terms.
Working with these expressions requires understanding different strategies for combining like terms, factoring, and using distributive properties. Recognizing special patterns, such as perfect squares and cubes, can also simplify the process.
Remember, getting comfortable with algebraic expressions allows you to set the stage for successfully solving equations and exploring more complex math concepts.

The Role of the Quadratic Formula in Solving Equations

The quadratic formula is a powerful tool used to find the roots of any quadratic equation. This formula is \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
The term \( b^2 - 4ac \) is called the discriminant and it provides vital information about the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one root, often referred to as a repeated root.
• If negative, the quadratic has complex roots, meaning they are not real numbers.

In the exercise, after simplifying the equation, we end up with a quadratic equation. We used the quadratic formula to find the potential solutions for \( m \). Evaluating the discriminant reveals important clues about the nature of these solutions.

Exploring Cube Roots

Cube roots are the inverse operation of cubing a number. In mathematical notation, the cube root of a number \( x \) is expressed as \( x^{1/3} \). This operation is central to equations that involve expressions like \( (m^3 + 26)^{1/3} \).
Computing a cube root can be tricky, as it involves finding a number that, when multiplied by itself three times, results in the original number.
Here's why cube roots are noteworthy:

• Unlike square roots, a cube root can result in negative outcomes due to the nature of multiplication in threes. For example, \((-8)^{1/3} = -2\) because \((-2) \times (-2) \times (-2) = -8\).
• In equations, cube roots can help simplify expressions with power terms, acting as a balancing force to eliminate the exponentiation.

Mastering cube roots enables students to manage more complex mathematical problems, making it easier to solve equations and understand the properties of numbers.