Question

Solve each equation. $$ 3^{x^{3}}=9^{x} $$

Short answer

The solutions to the equation are x = 0, x = √2, and x = -√2.

Step-by-step solution

Rewrite the equation using the same base

First, notice that both 3 and 9 are powers of 3. Recall that 9 can be written as a power of 3: 9 = 3^2. Therefore, rewrite the original equation using base 3: 3^{x^3} = (3^2)^x

Simplify the exponents

Apply the power rule (a^m)^n = a^{mn} to the right-hand side of the equation: 3^{x^3} = 3^{2x}

Set the exponents equal to each other

Since the bases are the same, the exponents must be equal: x^3 = 2x

Solve the equation for x

First, move all terms to one side: x^3 - 2x = 0 Factor out the common factor, x: x(x^2 - 2) = 0 This gives us two equations to solve: x = 0 and x^2 - 2 = 0

Solve the quadratic equation

Solve the quadratic equation x^2 - 2 = 0 by adding 2 to both sides: x^2 = 2 Take the square root of both sides to solve for x: x = ±√2

Key concepts

Solving Equations

To solve an equation, we are looking for the value(s) of the variable that make the equation true. For example, if we have the equation \(3^{x^3} = 9^x\), our goal is to find the value(s) of \(x\).
When solving exponential equations, it's helpful to rewrite each side of the equation so they have the same base if possible. This makes it easier to compare and solve the problem. Here we can see that 3 and 9 share a base. Specifically, 9 is equal to \(3^2\). Once the equations are rewritten with the same base, we can equate the exponents and solve for the variable.
Always keep in mind to simplify your equation step by step, ensuring clarity and accuracy in each transformation.

Factoring

Factoring involves breaking down an expression into a product of simpler expressions. This is a powerful tool in solving equations. For example, once we have the simplified equation from the problem \(x^3 - 2x = 0\), we can factor out the common term \(x\).
This becomes \(x(x^2 - 2) = 0\). Now, it's easier to solve because the product is zero only if at least one of the factors is zero. So, we set each factor equal to zero:

• \(x = 0\)
• \(x^2 - 2 = 0\)

By factoring, we decompose complex problems into simpler ones that we can solve more easily.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable \(x\) with the general form \(ax^2 + bx + c = 0\). In our problem, we are left with the quadratic equation \(x^2 - 2 = 0\) after factoring.
Solving quadratic equations can be done in several ways:

• Factoring
• Using the quadratic formula
• Completing the square

In this example, solving \(x^2 - 2 = 0\) is straightforward, as it's already in an easily solvable form. By adding 2 to both sides, we get \(x^2 = 2\). Taking the square root of both sides gives us \(x = ±\text{√}2\). These are the solutions to the quadratic part of our original equation.

Exponent Rules

Exponents are a fundamental concept in algebra, and understanding their rules simplifies solving exponential equations. One key rule is the power of a power rule: \((a^m)^n = a^{mn}\). This helps in rewriting exponential equations with the same base.
For instance, knowing that \(9 = 3^2\), we can rewrite the original equation \(3^{x^3} = 9^x\) as \(3^{x^3} = (3^2)^x\). Applying the power of a power rule transforms it into \(3^{x^3} = 3^{2x}\).
Once the bases are the same, we can set the exponents equal to each other. This reduces the problem to solving a simpler equation. Mastery of exponent rules is crucial for efficiently tackling and solving exponential problems.