Question
21) \( \frac{4^{2 x-1}}{4^{2 x}}=4^{3 x} \)
Ask by Marshall Barker. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = -\frac{1}{3} \)
Solution
To solve the equation:
\[
\frac{4^{2x-1}}{4^{2x}} = 4^{3x}
\]
**Step 1: Simplify the Left Side**
Using the properties of exponents, specifically \(\frac{a^m}{a^n} = a^{m-n}\), we simplify the left side:
\[
\frac{4^{2x-1}}{4^{2x}} = 4^{(2x-1) - 2x} = 4^{-1} = \frac{1}{4}
\]
So the equation becomes:
\[
\frac{1}{4} = 4^{3x}
\]
**Step 2: Express \(\frac{1}{4}\) with Base 4**
Notice that \(\frac{1}{4}\) can be written as \(4^{-1}\):
\[
4^{-1} = 4^{3x}
\]
**Step 3: Equate the Exponents**
Since the bases are the same and the equation \(4^a = 4^b\) implies \(a = b\), we set the exponents equal to each other:
\[
-1 = 3x
\]
**Step 4: Solve for \(x\)**
\[
x = \frac{-1}{3}
\]
**Final Answer:**
\[
x = -\frac{1}{3}
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( \frac{4^{2x-1}}{4^{2x}}=4^{3x} \), we can simplify the left side using the property of exponents: \[ \frac{4^{2x-1}}{4^{2x}} = 4^{(2x-1)-2x} = 4^{-1} = \frac{1}{4} \] Now the equation can be rewritten as: \[ \frac{1}{4} = 4^{3x} \] Since \( 4 = 4^1 \), we can express \( \frac{1}{4} \) as \( 4^{-1} \): \[ 4^{-1} = 4^{3x} \] Now we can set the exponents equal to each other because the bases are the same: \[ -1 = 3x \] Solving for \( x \) gives: \[ x = -\frac{1}{3} \] The solution to the equation is \( x = -\frac{1}{3} \).
