Question

Solve the equation. $$(4 n-6)^{3}=0$$

Short answer

The solution to the equation is \(n = 1.5\).

Step-by-step solution

Understand the equation

The given equation is: \((4n - 6)^3 = 0\) Only 0 to the power of any number will result in 0. Therefore, we can deduce that the expression inside the power, \(4n-6\), must be equal to 0.

Set the term inside the power equal to 0

Now, let's set \(4n - 6 = 0\) and solve for n.

Solve the equation

Add 6 to both sides of the equation to isolate the term with n: \(4n = 6\). Then, divide both sides by 4 to solve for n: \(n = 6/4 = 1.5\)

Key concepts

Algebraic Equations

An algebraic equation is a statement that two algebraic expressions are equal. It consists of variables, coefficients, and constants, along with algebraic operations such as addition, subtraction, multiplication, and division. In the context of our exercise, \( (4n - 6)^3 = 0 \) is an algebraic equation where \( (4n - 6) \) is raised to the 3rd power. To solve it, you need to first understand the implications of the equation as a whole. The result of raising any real number to the third power and achieving zero suggests that the base of the exponent—here, the expression \(4n - 6\)\—must equal zero. Identifying this fundamental property of the equation is critical, as it sets the stage for the solution process, which will involve isolating the variable to find its value.

Exponents and Powers

Exponents and powers refer to the number of times a number, called the base, is multiplied by itself. An exponent is displayed as a superscript to the right of the base. In our exercise, the expression \( (4n - 6)^3 \) involves the base \(4n - 6\)\ being raised to the power of 3. This means you would normally multiply \(4n - 6\)\ by itself three times. However, if the result of this exponentiation is zero, the base must be zero, as anything other than zero raised to a power does not equal zero. This characteristic is fundamental to understanding how to approach solving the cubic equation at hand—and underscores the power of exponents in shaping equations.

Isolate Variable

To isolate a variable means to rearrange an algebraic equation so that the variable appears on one side of the equation by itself. This process often involves performing inverse operations—such as adding opposites or dividing by a coefficient—on both sides of the equation. In our step by step solution, we isolated the variable n from the equation \(4n - 6 = 0\)\ by first adding 6 to both sides of the equation, eliminating the -6, and then dividing both sides by 4 to cancel the coefficient of n, thereby obtaining the solution \(n = 1.5\)\. These operations are crucial to solve for the variable and to derive the solution to the algebraic equation. It's important to perform the same operation on both sides of the equation to maintain the equality, which is a fundamental principle in algebra.