Question
By what number should \( (-6)^{-1} \) be multiplid so that the product becomes \( 9^{-1} \)
Ask by Patel Murphy. in India
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The number to multiply \( (-6)^{-1} \) by to get \( 9^{-1} \) is \(-\frac{2}{3}\).
Solution
To find the number \( x \) such that:
\[
(-6)^{-1} \times x = 9^{-1}
\]
Let's solve for \( x \):
1. **Understand the Negative Exponents:**
\[
(-6)^{-1} = \frac{1}{-6} = -\frac{1}{6}
\]
\[
9^{-1} = \frac{1}{9}
\]
2. **Set Up the Equation:**
\[
-\frac{1}{6} \times x = \frac{1}{9}
\]
3. **Solve for \( x \):**
Multiply both sides by \( -6 \) to isolate \( x \):
\[
x = \frac{1}{9} \times (-6) = -\frac{6}{9} = -\frac{2}{3}
\]
**Therefore, the number \( x \) is \(-\frac{2}{3}\).**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the problem, we need to find the number \( x \) such that: \[ (-6)^{-1} \cdot x = 9^{-1}. \] First, let's simplify \( (-6)^{-1} \) and \( 9^{-1} \): \[ (-6)^{-1} = \frac{1}{-6} = -\frac{1}{6}, \] \[ 9^{-1} = \frac{1}{9}. \] Now we can set up the equation: \[ -\frac{1}{6} \cdot x = \frac{1}{9}. \] To isolate \( x \), we can multiply both sides of the equation by \(-6\): \[ x = \frac{1}{9} \cdot (-6). \] Calculating this gives: \[ x = -\frac{6}{9} = -\frac{2}{3}. \] Thus, the number by which \( (-6)^{-1} \) should be multiplied to yield \( 9^{-1} \) is \[ \boxed{-\frac{2}{3}}. \]
