10x2y/7z2t6*(-35z6t3/18xy)
To simplify the expression \( \frac{10x^2y}{7z^2t^6} \times \left( \frac{-35z^6t^3}{18xy} \right) \), we can follow these steps:
Multiply the numerators and the denominators:
\[ \frac{10x^2y \cdot (-35z^6t^3)}{7z^2t^6 \cdot 18xy} \]
Multiply the numbers and variables in the numerator:
In the numerator:
- Coefficients: \( 10 \cdot (-35) = -350 \)
- For \( x \): \( x^2 \cdot x^1 = x^{2 + 1} = x^3 \) (since \( y \) is \( y^1 \))
- For \( y \): \( y^1 \)
- For \( z \): \( z^6 \)
- For \( t \): \( t^3 \)
So, the numerator becomes: \[ -350x^3y z^6 t^3 \]
Multiply the numbers and variables in the denominator:
In the denominator:
- Coefficients: \( 7 \cdot 18 = 126 \)
- For \( x \): \( x^1 \)
- For \( y \): \( y^1 \)
- For \( z \): \( z^2 \)
- For \( t \): \( t^6 \)
So, the denominator becomes: \[ 126xyz^2t^6 \]
Combine the results:
Putting it all together, we get: \[ \frac{-350x^3y z^6 t^3}{126xyz^2t^6} \]
Simplify the expression:
Coefficients: \[ \frac{-350}{126} = -\frac{175}{63} = -\frac{25}{9} \text{ (by dividing both by 7)} \]
For \( x \): \[ \frac{x^3}{x^1} = x^{3-1} = x^2 \]
For \( y \): \[ \frac{y^1}{y^1} = y^{1-1} = y^0 = 1 \]
For \( z \): \[ \frac{z^6}{z^2} = z^{6-2} = z^4 \]
For \( t \): \[ \frac{t^3}{t^6} = t^{3-6} = t^{-3} = \frac{1}{t^3} \]
Thus, combining everything: \[ \frac{-25}{9} \cdot x^2 \cdot z^4 \cdot \frac{1}{t^3} = -\frac{25x^2z^4}{9t^3} \]
The final simplified expression is:
\[ -\frac{25x^2z^4}{9t^3} \]
