Problem 59 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ 7^{a}=49^{-4} $$

Step-by-Step Solution

Answer

The solution is \(a = -8\).

Step 1: Express in Same Base

Recognize that both sides of the equation can be expressed as powers of 7. Since \(49 = 7^2\), we can write \(49^{-4}\) as \((7^2)^{-4}\).

Step 2: Simplify the Right Side

Apply the power of a power property, which states \((a^m)^n = a^{m \cdot n}\). So, \((7^2)^{-4} = 7^{2 \cdot (-4)} = 7^{-8}\).

Step 3: Equate Exponents

Now that both sides are expressed with the same base, equate the exponents: \[a = -8\]

Step 4: Verify the Solution

Substitute \(a = -8\) back into the original equation to check: \[7^{-8} = 49^{-4}\]Since both sides simplify to the same value, the solution is verified.

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. They often involve exponential expressions with the same base but different exponents on each side of the equation. The goal is to find the unknown value of the exponent. In this type of problem, identifying a common base can simplify the process significantly.
One common strategy for solving exponential equations is converting all terms to have the same base. For example, if you have two expressions like \(7^a\) and \(49^{-4}\), you can recognize that both numbers (7 and 49) can be related if expressed as powers of the same base. Identifying this is usually the first step towards solving such equations.

Solving Equations

Solving equations is a fundamental skill in algebra, and it involves finding the value of an unknown variable that makes the equation true. In the context of exponential equations, once you have expressed all terms with the same base, the equation transforms from a complex exponential form to a simpler algebraic form.

• Reduce the equation to have the same base on both sides by recognizing numbers that can be expressed with a common base.
• Apply properties and rules such as the power of a power property to simplify the terms.
• Equate the exponents once the bases are the same, allowing you to solve for the unknown variable using basic algebra.

This process reduces the complexity of exponential equations, making it easier to find solutions.

Power of a Power Property

The power of a power property is an essential rule when working with exponentials. This property states that when you raise a power to another power, you multiply the exponents together. Mathematically, it is expressed as \((a^m)^n = a^{m \cdot n}\). This property helps simplify exponential expressions.
In our problem, the right side of the equation was \((7^2)^{-4}\). Applying the power of a power property simplifies this expression to \(7^{2 \cdot (-4)} = 7^{-8}\). This simplification step is crucial as it allows both sides of the equation to be expressed with the same base, facilitating the solution of the given exponential equation.

• Identify when a power of a power is present in an equation.
• Apply the property to simplify expressions to their simplest form.
• This simplification helps in comparing and solving equations efficiently.