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Chapter 12: Problem 98

Solve. If no solution exists, state this. $$ 3^{3 x} \cdot 3^{x^{2}}=81 $$

Short Answer

Expert verified
The solutions are \( x = -4 \) and \( x = 1 \).

Step by step solution

01

Simplify the left side

Combine the exponents on the left side using the product rule for exponents: \[ 3^{3x} \times 3^{x^2} = 3^{3x + x^2} \] This simplifies the left-hand side to a single exponent.
02

Express the right side with the same base

Rewrite 81 as a power of 3. Since \[ 81 = 3^4, \] we can substitute 3^4 for 81 on the right side of the equation.
03

Set the exponents equal

Since the bases are the same, set the exponents equal to each other: \[ 3x + x^2 = 4. \]
04

Solve the quadratic equation

Rearrange the equation to a standard quadratic form: \[ x^2 + 3x - 4 = 0. \] Factor the quadratic: \[ (x + 4)(x - 1) = 0. \]
05

Find the solutions

Set each factor equal to zero and solve for x: \[ x + 4 = 0 \rightarrow x = -4, \ x - 1 = 0 \rightarrow x = 1. \] The solutions are \ x = -4 \ and \ x = 1. \

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Key Concepts

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

Product Rule for Exponents
When you see an expression involving exponents being multiplied, like in the exercise: \[ 3^{3x} \times 3^{x^2},\] you can use the product rule for exponents for simplification. The product rule says that when you multiply two expressions with the same base, you can add their exponents. So, \[ a^m \times a^n = a^{m+n}.\] This means we can rewrite \[ 3^{3x} \times 3^{x^2}\] as \[ 3^{3x + x^2}\] because the base (3) is the same. This step simplifies the expression to a single exponent.
Quadratic Equations
Quadratic equations are a type of polynomial equation of the form \[ ax^2 + bx + c = 0,\] where \( a, b, and \) and \( c \) are constants. In our exercise, after applying the product rule and rewriting 81 as a power of 3, we get: \[3x + x^2 = 4.\] To solve this as a quadratic equation, we rearrange it into the standard form:\

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