Problem 524 - Precalculus

Question

Solve \(\frac{125}{\left(\frac{1}{625}\right)^{-x-3}}=5^{3}\) by rewriting each side with a common base.

Step-by-Step Solution

Answer

The solution is \( x = -\frac{9}{4} \).

Step 1: Express both sides with base 5

First, express 125 and 625 as powers of the common base 5. We have 125 as a power of 5 which is written as \(5^3\), and 625 is written as \(5^4\). Thus, we rewrite the equation as:\[\frac{5^3}{\left(\frac{1}{5^4}\right)^{-x-3}} = 5^3\]

Step 2: Simplify the denominator

The denominator \(\left(\frac{1}{5^4}\right)^{-x-3}\) simplifies by using the rule \(\left(a^m\right)^n = a^{m \cdot n}\). So, we have:\[\left(5^{-4}\right)^{-x-3} = 5^{4(x+3)}\]

Step 3: Substitute simplified expressions into the equation

Now substituting \(5^{4(x+3)}\) into the initial equation, our equation becomes:\[\frac{5^3}{5^{4(x+3)}} = 5^3\]

Step 4: Simplify the fraction

Apply the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\). Therefore, simplify:\[5^{3-4(x+3)} = 5^3\]

Step 5: Set exponents equal

Since the bases are identical, the exponents must be equal. Thus, set the exponents equal to each other:\[3 - 4(x+3) = 3\]

Step 6: Solve the equation for x

Simplify the equation from the previous step:\[3 - 4x - 12 = 3\]Combine like terms and solve for \(x\):\[-4x - 9 = 0 \-4x = 9 \x = -\frac{9}{4}\]

Step 7: Final Step: Verify the solution

Substitute \(x = -\frac{9}{4}\) back into the original expression to check correctness. Express \(5^{4(-\frac{9}{4}+3)} = 5^3\). The solution holds if both sides equal 5^3, confirming \(x = -\frac{9}{4}\) is correct.

Key Concepts

Common Base

In mathematics, especially when working with exponential equations, rewriting expressions using a common base can simplify problem-solving significantly. The core idea is to express all terms of an equation as powers of the same base.

This approach allows you to manipulate the exponents more easily. For example, if you're given an equation where different numbers are involved, try breaking them down into powers of a small prime number, like 2 or 5.

An illustrative example involves the numbers 125 and 625. To rewrite them as powers of a common base (in this case, 5), you note that:

• 125 is equivalent to \(5^3\)
• 625 can be expressed as \(5^4\)

When the equation is rewritten using this common base, comparing and manipulating the exponents becomes straightforward, making it easier to solve the equation.

Properties of Exponents

The properties of exponents are fundamental rules that allow us to simplify and solve equations involving exponential terms. These properties include:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)

Understanding these properties is crucial for manipulating exponents efficiently. For instance, in the context of our exercise, the denominator \(\left(\frac{1}{5^4}\right)^{-x-3}\) can be simplified to \(5^{4(x+3)}\) using the power of a power property.

This rewriting allows us to consolidate the expression and eventually set the exponents equal when the bases are the same on both sides of the equation.

Solving Equations

Solving equations, particularly those involving exponents, requires applying algebraic techniques and the properties of exponents we've just discussed.

When both sides of an equation have the same base, like \(5^3\) and \(5^{3-4(x+3)}\), you can equate the exponents and solve the resulting algebraic equation. The steps include:

• Setting the exponents equal: \(3 - 4(x+3) = 3\)
• Simplifying the expression: \(3 - 4x - 12 = 3\)
• Solving for x by isolating the variable: \(-4x = 9\)
• Finding the value of x: \(x = -\frac{9}{4}\)

This process involves basic algebraic manipulation such as combining like terms and isolating the variable on one side. By translating the problem into a familiar algebraic structure, it becomes much more straightforward to find a solution. Verifying the solution by plugging it back into the original expression ensures it satisfies the equation.