Solve Using identity find product of ((ba)/(2b) + (3b)/4)(a/(2b) - (3b)/4) | 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

$$\frac { ( \frac { 4 } { 25 } + \frac { 3 b } { 4 } ) ( \frac { 9 } { 2 b } - \frac { 3 b } { 4 } ) } { ( \frac { a } { 2 b } - \frac { 3 b } { 4 } ) }$$

Evaluate

$\frac{3\left(75b+16\right)\left(6-b^{2}\right)}{100\left(2a-3b^{2}\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $25$ and $4$ is $100$. Multiply $\frac{4}{25}$ times $\frac{4}{4}$. Multiply $\frac{3b}{4}$ times $\frac{25}{25}$.

$$\frac{\left(\frac{4\times 4}{100}+\frac{25\times 3b}{100}\right)\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Since $\frac{4\times 4}{100}$ and $\frac{25\times 3b}{100}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\times 4+25\times 3b}{100}\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Do the multiplications in $4\times 4+25\times 3b$.

$$\frac{\frac{16+75b}{100}\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2b$ and $4$ is $4b$. Multiply $\frac{9}{2b}$ times $\frac{2}{2}$. Multiply $\frac{3b}{4}$ times $\frac{b}{b}$.

$$\frac{\frac{16+75b}{100}\left(\frac{9\times 2}{4b}-\frac{3bb}{4b}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Since $\frac{9\times 2}{4b}$ and $\frac{3bb}{4b}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{16+75b}{100}\times \frac{9\times 2-3bb}{4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

Do the multiplications in $9\times 2-3bb$.

$$\frac{\frac{16+75b}{100}\times \frac{18-3b^{2}}{4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

Multiply $\frac{16+75b}{100}$ times $\frac{18-3b^{2}}{4b}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

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

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a}{4b}-\frac{3bb}{4b}}$$

Since $\frac{2a}{4b}$ and $\frac{3bb}{4b}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a-3bb}{4b}}$$

Do the multiplications in $2a-3bb$.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a-3b^{2}}{4b}}$$

Divide $\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}$ by $\frac{2a-3b^{2}}{4b}$ by multiplying $\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}$ by the reciprocal of $\frac{2a-3b^{2}}{4b}$.

$$\frac{\left(16+75b\right)\left(18-3b^{2}\right)\times 4b}{100\times 4b\left(2a-3b^{2}\right)}$$

Cancel out $4b$ in both numerator and denominator.

$$\frac{\left(75b+16\right)\left(-3b^{2}+18\right)}{4\times 25\left(2a-3b^{2}\right)}$$

Multiply $4$ and $25$ to get $100$.

$$\frac{\left(75b+16\right)\left(-3b^{2}+18\right)}{100\left(2a-3b^{2}\right)}$$

Apply the distributive property by multiplying each term of $75b+16$ by each term of $-3b^{2}+18$.

$$\frac{-225b^{3}+1350b-48b^{2}+288}{100\left(2a-3b^{2}\right)}$$

Use the distributive property to multiply $100$ by $2a-3b^{2}$.

$$\frac{-225b^{3}+1350b-48b^{2}+288}{200a-300b^{2}}$$

Expand

$-\frac{3\left(75b^{3}+16b^{2}-450b-96\right)}{100\left(2a-3b^{2}\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $25$ and $4$ is $100$. Multiply $\frac{4}{25}$ times $\frac{4}{4}$. Multiply $\frac{3b}{4}$ times $\frac{25}{25}$.

$$\frac{\left(\frac{4\times 4}{100}+\frac{25\times 3b}{100}\right)\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Since $\frac{4\times 4}{100}$ and $\frac{25\times 3b}{100}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\times 4+25\times 3b}{100}\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Do the multiplications in $4\times 4+25\times 3b$.

$$\frac{\frac{16+75b}{100}\left(\frac{9}{2b}-\frac{3b}{4}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2b$ and $4$ is $4b$. Multiply $\frac{9}{2b}$ times $\frac{2}{2}$. Multiply $\frac{3b}{4}$ times $\frac{b}{b}$.

$$\frac{\frac{16+75b}{100}\left(\frac{9\times 2}{4b}-\frac{3bb}{4b}\right)}{\frac{a}{2b}-\frac{3b}{4}}$$

Since $\frac{9\times 2}{4b}$ and $\frac{3bb}{4b}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{16+75b}{100}\times \frac{9\times 2-3bb}{4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

Do the multiplications in $9\times 2-3bb$.

$$\frac{\frac{16+75b}{100}\times \frac{18-3b^{2}}{4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

Multiply $\frac{16+75b}{100}$ times $\frac{18-3b^{2}}{4b}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{a}{2b}-\frac{3b}{4}}$$

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

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a}{4b}-\frac{3bb}{4b}}$$

Since $\frac{2a}{4b}$ and $\frac{3bb}{4b}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a-3bb}{4b}}$$

Do the multiplications in $2a-3bb$.

$$\frac{\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}}{\frac{2a-3b^{2}}{4b}}$$

Divide $\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}$ by $\frac{2a-3b^{2}}{4b}$ by multiplying $\frac{\left(16+75b\right)\left(18-3b^{2}\right)}{100\times 4b}$ by the reciprocal of $\frac{2a-3b^{2}}{4b}$.

$$\frac{\left(16+75b\right)\left(18-3b^{2}\right)\times 4b}{100\times 4b\left(2a-3b^{2}\right)}$$

Cancel out $4b$ in both numerator and denominator.

$$\frac{\left(75b+16\right)\left(-3b^{2}+18\right)}{4\times 25\left(2a-3b^{2}\right)}$$

Multiply $4$ and $25$ to get $100$.

$$\frac{\left(75b+16\right)\left(-3b^{2}+18\right)}{100\left(2a-3b^{2}\right)}$$

Apply the distributive property by multiplying each term of $75b+16$ by each term of $-3b^{2}+18$.

$$\frac{-225b^{3}+1350b-48b^{2}+288}{100\left(2a-3b^{2}\right)}$$

Use the distributive property to multiply $100$ by $2a-3b^{2}$.

$$\frac{-225b^{3}+1350b-48b^{2}+288}{200a-300b^{2}}$$