Solve ((2b)/(b - 4) - 3/(b ^ 2))/(5/(5b - 20) + 3/(4b ^ 2 - 16b)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{\frac{2b}{b-4}-\frac{3}{b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Evaluate

$\frac{4\left(2b^{3}-3b+12\right)}{b\left(4b+3\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $b-4$ and $b^{2}$ is $\left(b-4\right)b^{2}$. Multiply $\frac{2b}{b-4}$ times $\frac{b^{2}}{b^{2}}$. Multiply $\frac{3}{b^{2}}$ times $\frac{b-4}{b-4}$.

$$\frac{\frac{2bb^{2}}{\left(b-4\right)b^{2}}-\frac{3\left(b-4\right)}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Since $\frac{2bb^{2}}{\left(b-4\right)b^{2}}$ and $\frac{3\left(b-4\right)}{\left(b-4\right)b^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2bb^{2}-3\left(b-4\right)}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Do the multiplications in $2bb^{2}-3\left(b-4\right)$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Factor $5b-20$. Factor $4b^{2}-16b$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5}{5\left(b-4\right)}+\frac{3}{4b\left(b-4\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $5\left(b-4\right)$ and $4b\left(b-4\right)$ is $20b\left(b-4\right)$. Multiply $\frac{5}{5\left(b-4\right)}$ times $\frac{4b}{4b}$. Multiply $\frac{3}{4b\left(b-4\right)}$ times $\frac{5}{5}$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\times 4b}{20b\left(b-4\right)}+\frac{3\times 5}{20b\left(b-4\right)}}$$

Since $\frac{5\times 4b}{20b\left(b-4\right)}$ and $\frac{3\times 5}{20b\left(b-4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\times 4b+3\times 5}{20b\left(b-4\right)}}$$

Do the multiplications in $5\times 4b+3\times 5$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{20b+15}{20b\left(b-4\right)}}$$

Factor the expressions that are not already factored in $\frac{20b+15}{20b\left(b-4\right)}$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\left(4b+3\right)}{20b\left(b-4\right)}}$$

Cancel out $5$ in both numerator and denominator.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{4b+3}{4b\left(b-4\right)}}$$

Divide $\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}$ by $\frac{4b+3}{4b\left(b-4\right)}$ by multiplying $\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}$ by the reciprocal of $\frac{4b+3}{4b\left(b-4\right)}$.

$$\frac{\left(2b^{3}-3b+12\right)\times 4b\left(b-4\right)}{\left(b-4\right)b^{2}\left(4b+3\right)}$$

Cancel out $b\left(b-4\right)$ in both numerator and denominator.

$$\frac{4\left(2b^{3}-3b+12\right)}{b\left(4b+3\right)}$$

Use the distributive property to multiply $4$ by $2b^{3}-3b+12$.

$$\frac{8b^{3}-12b+48}{b\left(4b+3\right)}$$

Use the distributive property to multiply $b$ by $4b+3$.

$$\frac{8b^{3}-12b+48}{4b^{2}+3b}$$

Expand

$\frac{4\left(2b^{3}-3b+12\right)}{b\left(4b+3\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $b-4$ and $b^{2}$ is $\left(b-4\right)b^{2}$. Multiply $\frac{2b}{b-4}$ times $\frac{b^{2}}{b^{2}}$. Multiply $\frac{3}{b^{2}}$ times $\frac{b-4}{b-4}$.

$$\frac{\frac{2bb^{2}}{\left(b-4\right)b^{2}}-\frac{3\left(b-4\right)}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Since $\frac{2bb^{2}}{\left(b-4\right)b^{2}}$ and $\frac{3\left(b-4\right)}{\left(b-4\right)b^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2bb^{2}-3\left(b-4\right)}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Do the multiplications in $2bb^{2}-3\left(b-4\right)$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5}{5b-20}+\frac{3}{4b^{2}-16b}}$$

Factor $5b-20$. Factor $4b^{2}-16b$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5}{5\left(b-4\right)}+\frac{3}{4b\left(b-4\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $5\left(b-4\right)$ and $4b\left(b-4\right)$ is $20b\left(b-4\right)$. Multiply $\frac{5}{5\left(b-4\right)}$ times $\frac{4b}{4b}$. Multiply $\frac{3}{4b\left(b-4\right)}$ times $\frac{5}{5}$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\times 4b}{20b\left(b-4\right)}+\frac{3\times 5}{20b\left(b-4\right)}}$$

Since $\frac{5\times 4b}{20b\left(b-4\right)}$ and $\frac{3\times 5}{20b\left(b-4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\times 4b+3\times 5}{20b\left(b-4\right)}}$$

Do the multiplications in $5\times 4b+3\times 5$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{20b+15}{20b\left(b-4\right)}}$$

Factor the expressions that are not already factored in $\frac{20b+15}{20b\left(b-4\right)}$.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{5\left(4b+3\right)}{20b\left(b-4\right)}}$$

Cancel out $5$ in both numerator and denominator.

$$\frac{\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}}{\frac{4b+3}{4b\left(b-4\right)}}$$

Divide $\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}$ by $\frac{4b+3}{4b\left(b-4\right)}$ by multiplying $\frac{2b^{3}-3b+12}{\left(b-4\right)b^{2}}$ by the reciprocal of $\frac{4b+3}{4b\left(b-4\right)}$.

$$\frac{\left(2b^{3}-3b+12\right)\times 4b\left(b-4\right)}{\left(b-4\right)b^{2}\left(4b+3\right)}$$

Cancel out $b\left(b-4\right)$ in both numerator and denominator.

$$\frac{4\left(2b^{3}-3b+12\right)}{b\left(4b+3\right)}$$

Use the distributive property to multiply $4$ by $2b^{3}-3b+12$.

$$\frac{8b^{3}-12b+48}{b\left(4b+3\right)}$$

Use the distributive property to multiply $b$ by $4b+3$.

$$\frac{8b^{3}-12b+48}{4b^{2}+3b}$$