Solve 24m^3-2m/1-2msqrt{3}

$$\frac { 24 m ^ { 3 } - 2 m } { 1 - 2 m \sqrt { 3 } }$$

Evaluate

$$\frac{2m\left(12m^{2}-1\right)}{-2\sqrt{3}m+1}$$

Short Solution Steps

Multiply -1 and 2 to get -2.

$$\frac{24m^{3}-2m}{1-2m\sqrt{3}}$$

Rationalize the denominator of \frac{24m^{3}-2m}{1-2m\sqrt{3}} by multiplying numerator and denominator by 1+2m\sqrt{3}.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{\left(1-2m\sqrt{3}\right)\left(1+2m\sqrt{3}\right)}$$

Consider \left(1-2m\sqrt{3}\right)\left(1+2m\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1^{2}-\left(-2m\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-\left(-2m\sqrt{3}\right)^{2}}$$

Expand \left(-2m\sqrt{3}\right)^{2}.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-\left(-2\right)^{2}m^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-4m^{2}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-4m^{2}\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-12m^{2}}$$

Factor the expressions that are not already factored.

$$\frac{2m\left(2\sqrt{3}m+1\right)\left(12m^{2}-1\right)}{-12m^{2}+1}$$

Extract the negative sign in 12m^{2}-1.

$$\frac{-2m\left(2\sqrt{3}m+1\right)\left(-12m^{2}+1\right)}{-12m^{2}+1}$$

Cancel out -12m^{2}+1 in both numerator and denominator.

$$-2m\left(2\sqrt{3}m+1\right)$$

Expand the expression.

$$-4\sqrt{3}m^{2}-2m$$

Factor

$$\frac{2m\left(12m^{2}-1\right)}{-2\sqrt{3}m+1}$$

Solution Steps

Multiply -1 and 2 to get -2.

$$factor(\frac{24m^{3}-2m}{1-2m\sqrt{3}})$$

Rationalize the denominator of \frac{24m^{3}-2m}{1-2m\sqrt{3}} by multiplying numerator and denominator by 1+2m\sqrt{3}.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{\left(1-2m\sqrt{3}\right)\left(1+2m\sqrt{3}\right)})$$

Consider \left(1-2m\sqrt{3}\right)\left(1+2m\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1^{2}-\left(-2m\sqrt{3}\right)^{2}})$$

Calculate 1 to the power of 2 and get 1.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-\left(-2m\sqrt{3}\right)^{2}})$$

Expand \left(-2m\sqrt{3}\right)^{2}.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-\left(-2\right)^{2}m^{2}\left(\sqrt{3}\right)^{2}})$$

Calculate -2 to the power of 2 and get 4.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-4m^{2}\left(\sqrt{3}\right)^{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-4m^{2}\times 3})$$

Multiply 4 and 3 to get 12.

$$factor(\frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-12m^{2}})$$

Factor the expressions that are not already factored in \frac{\left(24m^{3}-2m\right)\left(1+2m\sqrt{3}\right)}{1-12m^{2}}.

$$factor(\frac{2m\left(2\sqrt{3}m+1\right)\left(12m^{2}-1\right)}{-12m^{2}+1})$$

Extract the negative sign in 12m^{2}-1.

$$factor(\frac{-2m\left(2\sqrt{3}m+1\right)\left(-12m^{2}+1\right)}{-12m^{2}+1})$$

Cancel out -12m^{2}+1 in both numerator and denominator.

$$factor(-2m\left(2\sqrt{3}m+1\right))$$

Expand the expression.

$$factor(-4\sqrt{3}m^{2}-2m)$$

Factor out 2.

$$2\left(-2\sqrt{3}m^{2}-m\right)$$

Consider -2\sqrt{3}m^{2}-m. Factor out m.

$$m\left(-2\sqrt{3}m-1\right)$$

Rewrite the complete factored expression.

$$2m\left(-2\sqrt{3}m-1\right)$$