Solve (y + 3)/(y ^ 2 + 6y + 9) + 6/(y ^ 2 + 9) - (16y)/(8y ^ 2 * 24y) | 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

$$\frac{y+3}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{2}24y}$$

Evaluate

$\frac{12y^{4}+71y^{3}+321y^{2}-9y-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add $2$ and $1$ to get $3$.

$$\frac{y+3}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Factor the expressions that are not already factored in $\frac{y+3}{y^{2}+6y+9}$.

$$\frac{y+3}{\left(y+3\right)^{2}}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Cancel out $y+3$ in both numerator and denominator.

$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Cancel out $2\times 8y$ in both numerator and denominator.

$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{1}{12y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $y+3$ and $y^{2}+9$ is $\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{1}{y+3}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{y+3}{y+3}$.

$$\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}+\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Since $\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{y^{2}+9+6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Do the multiplications in $y^{2}+9+6\left(y+3\right)$.

$$\frac{y^{2}+9+6y+18}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Combine like terms in $y^{2}+9+6y+18$.

$$\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)\left(y^{2}+9\right)$ and $12y^{2}$ is $12\left(y+3\right)y^{2}\left(y^{2}+9\right)$. Multiply $\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}$ times $\frac{12y^{2}}{12y^{2}}$. Multiply $\frac{1}{12y^{2}}$ times $\frac{\left(y+3\right)\left(y^{2}+9\right)}{\left(y+3\right)\left(y^{2}+9\right)}$.

$$\frac{\left(y^{2}+27+6y\right)\times 12y^{2}}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}-\frac{\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Since $\frac{\left(y^{2}+27+6y\right)\times 12y^{2}}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$ and $\frac{\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y^{2}+27+6y\right)\times 12y^{2}-\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Do the multiplications in $\left(y^{2}+27+6y\right)\times 12y^{2}-\left(y+3\right)\left(y^{2}+9\right)$.

$$\frac{12y^{4}+324y^{2}+72y^{3}-y^{3}-9y-3y^{2}-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Combine like terms in $12y^{4}+324y^{2}+72y^{3}-y^{3}-9y-3y^{2}-27$.

$$\frac{12y^{4}+321y^{2}+71y^{3}-9y-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Expand $12\left(y+3\right)y^{2}\left(y^{2}+9\right)$.

$$\frac{12y^{4}+321y^{2}+71y^{3}-9y-27}{12y^{5}+36y^{4}+108y^{3}+324y^{2}}$$

Expand

$\frac{12y^{4}+71y^{3}+321y^{2}-9y-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add $2$ and $1$ to get $3$.

$$\frac{y+3}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Factor the expressions that are not already factored in $\frac{y+3}{y^{2}+6y+9}$.

$$\frac{y+3}{\left(y+3\right)^{2}}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Cancel out $y+3$ in both numerator and denominator.

$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{3}\times 24}$$

Cancel out $2\times 8y$ in both numerator and denominator.

$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{1}{12y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $y+3$ and $y^{2}+9$ is $\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{1}{y+3}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{y+3}{y+3}$.

$$\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}+\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Since $\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{y^{2}+9+6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Do the multiplications in $y^{2}+9+6\left(y+3\right)$.

$$\frac{y^{2}+9+6y+18}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

Combine like terms in $y^{2}+9+6y+18$.

$$\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}-\frac{1}{12y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)\left(y^{2}+9\right)$ and $12y^{2}$ is $12\left(y+3\right)y^{2}\left(y^{2}+9\right)$. Multiply $\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}$ times $\frac{12y^{2}}{12y^{2}}$. Multiply $\frac{1}{12y^{2}}$ times $\frac{\left(y+3\right)\left(y^{2}+9\right)}{\left(y+3\right)\left(y^{2}+9\right)}$.

$$\frac{\left(y^{2}+27+6y\right)\times 12y^{2}}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}-\frac{\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Since $\frac{\left(y^{2}+27+6y\right)\times 12y^{2}}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$ and $\frac{\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y^{2}+27+6y\right)\times 12y^{2}-\left(y+3\right)\left(y^{2}+9\right)}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Do the multiplications in $\left(y^{2}+27+6y\right)\times 12y^{2}-\left(y+3\right)\left(y^{2}+9\right)$.

$$\frac{12y^{4}+324y^{2}+72y^{3}-y^{3}-9y-3y^{2}-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Combine like terms in $12y^{4}+324y^{2}+72y^{3}-y^{3}-9y-3y^{2}-27$.

$$\frac{12y^{4}+321y^{2}+71y^{3}-9y-27}{12\left(y+3\right)y^{2}\left(y^{2}+9\right)}$$

Expand $12\left(y+3\right)y^{2}\left(y^{2}+9\right)$.

$$\frac{12y^{4}+321y^{2}+71y^{3}-9y-27}{12y^{5}+36y^{4}+108y^{3}+324y^{2}}$$