Solve t (a ^ 2 - 6a + 9)/(a ^ 2 - 2a - 3) / ((a ^ 2 - 5a - 6)/(a ^ 2 - 3a + 2)) * ((a - 3) ^ 2)/((a - 1)(a - 2)) | 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

$$\frac { a ^ { 2 } - 6 a + 9 } { a ^ { 2 } - 2 a - 3 } \div \frac { a ^ { 2 } - 5 a - 6 } { a ^ { 2 } - 3 a + 2 } \times \frac { ( a - 3 ) ^ { 2 } } { ( a - 1 ) ( a - 2 ) }$$

Evaluate

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

Short Solution Steps

Divide $\frac{a^{2}-6a+9}{a^{2}-2a-3}$ by $\frac{a^{2}-5a-6}{a^{2}-3a+2}$ by multiplying $\frac{a^{2}-6a+9}{a^{2}-2a-3}$ by the reciprocal of $\frac{a^{2}-5a-6}{a^{2}-3a+2}$.

$$\frac{\left(a^{2}-6a+9\right)\left(a^{2}-3a+2\right)}{\left(a^{2}-2a-3\right)\left(a^{2}-5a-6\right)}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Factor the expressions that are not already factored in $\frac{\left(a^{2}-6a+9\right)\left(a^{2}-3a+2\right)}{\left(a^{2}-2a-3\right)\left(a^{2}-5a-6\right)}$.

$$\frac{\left(a-2\right)\left(a-1\right)\left(a-3\right)^{2}}{\left(a-6\right)\left(a-3\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Cancel out $a-3$ in both numerator and denominator.

$$\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Use the distributive property to multiply $a-1$ by $a-2$ and combine like terms.

$$\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{a^{2}-3a+2}$$

Multiply $\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}$ times $\frac{\left(a-3\right)^{2}}{a^{2}-3a+2}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out $\left(a-2\right)\left(a-1\right)$ in both numerator and denominator.

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

Expand the expression.

$$\frac{a^{3}-9a^{2}+27a-27}{a^{3}-4a^{2}-11a-6}$$

Expand

$\frac{a^{3}-9a^{2}+27a-27}{\left(a+1\right)\left(a^{2}-5a-6\right)}$

Short Solution Steps

Divide $\frac{a^{2}-6a+9}{a^{2}-2a-3}$ by $\frac{a^{2}-5a-6}{a^{2}-3a+2}$ by multiplying $\frac{a^{2}-6a+9}{a^{2}-2a-3}$ by the reciprocal of $\frac{a^{2}-5a-6}{a^{2}-3a+2}$.

$$\frac{\left(a^{2}-6a+9\right)\left(a^{2}-3a+2\right)}{\left(a^{2}-2a-3\right)\left(a^{2}-5a-6\right)}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Factor the expressions that are not already factored in $\frac{\left(a^{2}-6a+9\right)\left(a^{2}-3a+2\right)}{\left(a^{2}-2a-3\right)\left(a^{2}-5a-6\right)}$.

$$\frac{\left(a-2\right)\left(a-1\right)\left(a-3\right)^{2}}{\left(a-6\right)\left(a-3\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Cancel out $a-3$ in both numerator and denominator.

$$\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{\left(a-1\right)\left(a-2\right)}$$

Use the distributive property to multiply $a-1$ by $a-2$ and combine like terms.

$$\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}\times \frac{\left(a-3\right)^{2}}{a^{2}-3a+2}$$

Multiply $\frac{\left(a-3\right)\left(a-2\right)\left(a-1\right)}{\left(a-6\right)\left(a+1\right)^{2}}$ times $\frac{\left(a-3\right)^{2}}{a^{2}-3a+2}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out $\left(a-2\right)\left(a-1\right)$ in both numerator and denominator.

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

Expand the expression.

$$\frac{a^{3}-9a^{2}+27a-27}{a^{3}-4a^{2}-11a-6}$$