Problem 41 - Algebra 2

Question

Solve each equation. Check your solution. $$ \left(\frac{1}{9}\right)^{m}=81^{m+4} $$

Step-by-Step Solution

Answer

The solution to the equation is \(m = -\frac{8}{3}\).

Step 1: Rewrite Bases with Same Base Number

Notice that both sides of the equation involve powers of numbers that can be expressed in terms of a common base. Recall that 9 is the square of 3, and 81 is the fourth power of 3. Rewrite the equation as follows:\[\left(\frac{1}{3^2}\right)^m = \left(3^4\right)^{m+4}\]This can be simplified to:\[3^{-2m} = 3^{4(m+4)}\]

Step 2: Set Exponents Equal

Since the bases are the same, we equate the exponents:\[-2m = 4(m+4)\]

Step 3: Simplify the Equation

Expand the right side of the equation:\[-2m = 4m + 16\]

Step 4: Solve for m

Rearrange the equation to isolate \(m\):\[-2m - 4m = 16\]Combine like terms:\[-6m = 16\]Divide both sides by -6:\[m = \frac{16}{-6} = -\frac{8}{3}\]

Step 5: Check the Solution

Substitute \(m = -\frac{8}{3}\) back into the original equation to verify the solution. Calculate both sides to ensure they are equal:First, calculate the left side:\[\left(\frac{1}{9}\right)^{-\frac{8}{3}} = 9^{\frac{8}{3}} = (3^2)^{\frac{8}{3}} = 3^{\frac{16}{3}}\]Now, calculate the right side:\[81^{\left(-\frac{8}{3}+4\right)} = (3^4)^{\frac{4}{3}} = 3^{\frac{16}{3}}\]Both sides equal, confirming the solution is correct.

Key Concepts

Polynomial Equations

Polynomial equations are a type of equation that involves variables raised to whole number powers. Unlike linear equations which may have the variable terms to the first power only, polynomials can include quadratic terms, cubic terms, and so forth. For example, in the given problem, the intricate process of solving for \(m\) involves treating the equation as a polynomial equation once it is expressed in terms of a common base. Polynomial equations can often be solved by applying various algebraic techniques such as factoring or using specific formulas depending on the degree of the polynomial present in the equation.

Understanding the structure and properties of polynomial equations is crucial. This includes recognizing the coefficients, degrees, and the behavior of such equations. It gives insight into how the polynomials interact within different algebraic expressions and how solutions may change depending on the values substituted for the variables. This is particularly important when verifying solutions, as different values can modify the degree of the polynomial, impacting the solution significantly.

Exponent Rules

Exponent rules simplify complex expressions and are key in solving equations involving powers. They help in manipulating expressions to reach a solution more efficiently. In our exercise, we applied several exponent rules:

• Negative Exponent Rule: A negative exponent inverts the base, for example, \(a^{-n} = \frac{1}{a^n}\).
• Power of a Power Rule: When raising an exponent to another exponent, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Product of Powers Rule: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).

By changing bases while applying these rules, we simplify the given equation into a form that allows aligning the exponents, which are solved using polynomial arithmetic. Grasping these rules helps reduce complexity and opens paths to more simplified solutions. Regular practice will ensure these become second nature, enhancing problem-solving efficiency.

Base Conversion

In mathematics, base conversion is a technique used to change the base of expressions to facilitate solving equations. The goal is often to simplify or match bases on either side of an equation to use exponent rules effectively.

In the exercise, the base conversion was a critical step. Starting with \(\left( \frac{1}{9} \right)^m = 81^{m+4}\), both 9 and 81 were expressed in terms of the base of 3: since 9 is \(3^2\) and 81 is \(3^4\). After converting, the equation becomes \(3^{-2m} = 3^{4(m+4)}\).

This conversion is important because it allows the application of exponent rules to compare and set exponents equal. By converting bases to a common one, we reduce a complex problem into a simpler algebraic equation that is easier to handle mathematically, making it a powerful tool in an expert mathematician's toolkit.

Checking Solutions

Checking solutions is a step that verifies if the derived solution satisfies the original equation. This is crucial as complex algebraic manipulations can lead to mishaps or simplified forms that could differ from the actual problem's requirements.

To validate the solution \(m = -\frac{8}{3}\), it was substituted back into the initial equation. After substituting, both sides of the equation were calculated separately. The left side became \(9^{\frac{8}{3}}\) and the right side became \(3^{\frac{16}{3}}\). The calculations show that both sides equal \(3^{\frac{16}{3}}\).

This consistency signifies that the solution is correct. Checking solutions not only confirms accuracy but also enhances understanding. It encourages critical thinking and ensures that the solution aligns with the conditions set by the problem. Moreover, this practice helps uncover errors or offer insights for alternative approaches, fostering deeper learning and problem-solving confidence.