Problem 32 - Algebra A Combined Function

Question

Solve. $$ 125^{x-2}=25 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{8}{3}\).

Step 1: Express the Equation in Terms of Powers of 5

We start by expressing both sides of the equation as powers of 5. We know that:- 125 can be rewritten as \(5^3\), because \(125 = 5 \times 5 \times 5\).- 25 can be rewritten as \(5^2\), because \(25 = 5 \times 5\).Thus, the equation \(125^{x-2} = 25\) becomes \((5^3)^{x-2} = 5^2\).

Step 2: Apply the Power of a Power Property

Next, we apply the power of a power property \((a^m)^n = a^{m\cdot n}\) to the left side of the equation:\[(5^3)^{x-2} = 5^{3(x-2)}.\]

Step 3: Set the Exponents Equal to Each Other

Since the bases are now the same on both sides (both are 5), we can set the exponents equal to each other:\[3(x-2) = 2.\]

Step 4: Solve for x

We solve this equation for \(x\):- First, distribute the 3 on the left side: \(3x - 6 = 2.\)- Add 6 to both sides: \(3x = 8.\)- Divide both sides by 3: \(x = \frac{8}{3}.\)

Key Concepts

Powers of 5

Exponents simplify multiplication by indicating how many times a number, called the base, is multiplied by itself. For example, in the expression

• \(5^3\), we are multiplying 5 by itself 3 times: \(5 \times 5 \times 5 = 125\).
• Similarly, \(5^2\) means 5 is multiplied by itself 2 times: \(5 \times 5 = 25\).

Understanding how to express numbers as powers of common bases, like 5, is crucial in simplifying equations, especially those involving exponents.
By recognizing that 125 is \(5^3\) and 25 is \(5^2\), it becomes easier to compare and manipulate expressions with these powers. This skill is fundamental in solving equations where terms need to be expressed with the same base.

Power of a Power Property

The "Power of a Power" property in exponents is a handy rule that allows us to simplify expressions where an exponent is raised to another exponent. The rule states that

• \((a^m)^n = a^{m \cdot n}\).

This means that when you have an exponentiation inside another exponentiation, you can multiply the exponents together.
Let's consider the expression \((5^3)^{x-2}\). Using the power of a power property, we simplify this to \(5^{3(x-2)}\).
This step transforms the equation into a familiar form where only the exponents need to be considered, making it easier to isolate the variable and solve the equation.

Solving Equations with Exponents

Equations with exponents often require isolating the variable by using properties of exponents. Start by ensuring the bases on both sides of the equation are the same, as this allows you to compare the exponents directly.
In our example, we have

• \(5^{3(x-2)} = 5^2\).

Since the bases are both 5, we can equate the exponents:

• \(3(x-2) = 2\).

Solving this equation involves

• Distributing and simplifying: \(3x - 6 = 2\).
• Adding 6 to both sides: \(3x = 8\).
• Finally, dividing by 3: \(x = \frac{8}{3}\).

By focusing on manipulating exponents and the structure of the equation, you effectively solve for the unknown variable, which is key in handling more complex exponential equations.