Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( b = 3 \)
Solution
To solve the equation:
\[
10^{-2b - 3} = 10^{-3b}
\]
**Step 1: Recognize the Property of Exponents**
Since the bases are the same (both are 10) and greater than 0, we can set the exponents equal to each other:
\[
-2b - 3 = -3b
\]
**Step 2: Solve for \( b \)**
1. **Add \( 3b \) to both sides** to get all \( b \) terms on one side:
\[
-2b - 3 + 3b = -3b + 3b
\]
\[
b - 3 = 0
\]
2. **Add 3 to both sides** to isolate \( b \):
\[
b = 3
\]
**Solution:**
\[
b = 3
\]
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Simplify this solution Mind Expander
To solve the equation \( 10^{-2 b - 3} = 10^{-3 b} \), we can start by using the property of exponents that states if the bases are the same, we can set the exponents equal to each other. So, we have: \[ -2b - 3 = -3b \] Now, let's solve for \( b \). We'll add \( 3b \) to both sides: \[ -2b + 3b - 3 = 0 \] This simplifies to: \[ b - 3 = 0 \] Now, add 3 to both sides: \[ b = 3 \] Thus, the solution to the equation is \( b = 3 \).
