Solve (2(x)^(2))/(3y)\times(4(x)^(2))/(y)\div(8(x)^(3))/(9(y)^(2))

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Question

$$\frac{ 2 { x }^{ 2 } }{ 3y } \times \frac{ 4 { x }^{ 2 } }{ y } \div \frac{ 8 { x }^{ 3 } }{ 9 { y }^{ 2 } }$$

Evaluate

$3x$

Solution Steps

Multiply $\frac{2x^{2}}{3y}$ times $\frac{4x^{2}}{y}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2x^{2}\times 4x^{2}}{3yy}}{\frac{8x^{3}}{9y^{2}}}$$

Divide $\frac{2x^{2}\times 4x^{2}}{3yy}$ by $\frac{8x^{3}}{9y^{2}}$ by multiplying $\frac{2x^{2}\times 4x^{2}}{3yy}$ by the reciprocal of $\frac{8x^{3}}{9y^{2}}$.

$$\frac{2x^{2}\times 4x^{2}\times 9y^{2}}{3yy\times 8x^{3}}$$

Cancel out $2\times 3\times 4xyyx^{2}$ in both numerator and denominator.

$$3x$$

Show Solution

Differentiate w.r.t. x

$3$

Steps Using Definition of a Derivative

Multiply $\frac{2x^{2}}{3y}$ times $\frac{4x^{2}}{y}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{2x^{2}\times 4x^{2}}{3yy}}{\frac{8x^{3}}{9y^{2}}})$$

Divide $\frac{2x^{2}\times 4x^{2}}{3yy}$ by $\frac{8x^{3}}{9y^{2}}$ by multiplying $\frac{2x^{2}\times 4x^{2}}{3yy}$ by the reciprocal of $\frac{8x^{3}}{9y^{2}}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}\times 4x^{2}\times 9y^{2}}{3yy\times 8x^{3}})$$

Cancel out $2\times 3\times 4xyyx^{2}$ in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x)$$

The derivative of $ax^{n}$ is $nax^{n-1}$.

$$3x^{1-1}$$

Subtract $1$ from $1$.

$$3x^{0}$$

For any term $t$ except $0$, $t^{0}=1$.

$$3\times 1$$

For any term $t$, $t\times 1=t$ and $1t=t$.

$$3$$