Pre-Algebra Examples

$\frac{n^{5} + n^{2}}{n^{2} + n}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $n^{2}$ out of $n^{5} + n^{2}$ .

Tap for more steps...

Step 1.1.1

Factor $n^{2}$ out of $n^{5}$ .

$\frac{n^{2} n^{3} + n^{2}}{n^{2} + n}$

Step 1.1.2

Multiply by $1$ .

$\frac{n^{2} n^{3} + n^{2} \cdot 1}{n^{2} + n}$

Step 1.1.3

Factor $n^{2}$ out of $n^{2} n^{3} + n^{2} \cdot 1$ .

$\frac{n^{2} (n^{3} + 1)}{n^{2} + n}$

Step 1.2

Rewrite $1$ as $1^{3}$ .

$\frac{n^{2} (n^{3} + 1^{3})}{n^{2} + n}$

Step 1.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = n$ and $b = 1$ .

$\frac{n^{2} ((n + 1) (n^{2} - n \cdot 1 + 1^{2}))}{n^{2} + n}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Multiply $- 1$ by $1$ .

$\frac{n^{2} ((n + 1) (n^{2} - n + 1^{2}))}{n^{2} + n}$

Step 1.4.2

One to any power is one.

$\frac{n^{2} ((n + 1) (n^{2} - n + 1))}{n^{2} + n}$

$\frac{n^{2} (n + 1) (n^{2} - n + 1)}{n^{2} + n}$

Step 2

Factor $n$ out of $n^{2} + n$ .

Tap for more steps...

Step 2.1

Factor $n$ out of $n^{2}$ .

$\frac{n^{2} (n + 1) (n^{2} - n + 1)}{n \cdot n + n}$

Step 2.2

Raise $n$ to the power of $1$ .

$\frac{n^{2} (n + 1) (n^{2} - n + 1)}{n \cdot n + n^{1}}$

Step 2.3

Factor $n$ out of $n^{1}$ .

$\frac{n^{2} (n + 1) (n^{2} - n + 1)}{n \cdot n + n \cdot 1}$

Step 2.4

Factor $n$ out of $n \cdot n + n \cdot 1$ .

$\frac{n^{2} (n + 1) (n^{2} - n + 1)}{n (n + 1)}$

Step 3

Cancel the common factor of $n^{2}$ and $n$ .

Tap for more steps...

Step 3.1

Factor $n$ out of $n^{2} (n + 1) (n^{2} - n + 1)$ .

$\frac{n ((n (n + 1)) (n^{2} - n + 1))}{n (n + 1)}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$\frac{n ((n (n + 1)) (n^{2} - n + 1))}{n (n + 1)}$

Step 3.2.2

Rewrite the expression.

$\frac{(n (n + 1)) (n^{2} - n + 1)}{n + 1}$

Step 4

Cancel the common factor of $n + 1$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{n (n + 1) (n^{2} - n + 1)}{n + 1}$

Step 4.2

Divide $(n) (n^{2} - n + 1)$ by $1$ .

$(n) (n^{2} - n + 1)$

Step 5

Apply the distributive property.

$n \cdot n^{2} + n (- n) + n \cdot 1$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $n$ by $n^{2}$ by adding the exponents.

Tap for more steps...

Step 6.1.1

Multiply $n$ by $n^{2}$ .

Tap for more steps...

Step 6.1.1.1

Raise $n$ to the power of $1$ .

$n^{1} n^{2} + n (- n) + n \cdot 1$

Step 6.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$n^{1 + 2} + n (- n) + n \cdot 1$

Step 6.1.2

Add $1$ and $2$ .

$n^{3} + n (- n) + n \cdot 1$

Step 6.2

Rewrite using the commutative property of multiplication.

$n^{3} - n \cdot n + n \cdot 1$

Step 6.3

Multiply $n$ by $1$ .

$n^{3} - n \cdot n + n$

Step 7

Multiply $n$ by $n$ by adding the exponents.

Tap for more steps...

Step 7.1

Move $n$ .

$n^{3} - (n \cdot n) + n$

Step 7.2

Multiply $n$ by $n$ .

$n^{3} - n^{2} + n$