Algebra Examples

${(m + \frac{1}{m})}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(m + \frac{1}{m})}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $m$ and $b$ $\frac{1}{m}$ into the expression.

$1 {(m)}^{6} {(\frac{1}{m})}^{0} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(m)}^{6}$ by $1$ .

${(m)}^{6} {(\frac{1}{m})}^{0} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.2

Apply the product rule to $\frac{1}{m}$ .

$m^{6} \frac{1^{0}}{m^{0}} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.3

Anything raised to $0$ is $1$ .

$m^{6} \frac{1}{m^{0}} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.4

Anything raised to $0$ is $1$ .

$m^{6} \frac{1}{1} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.5

Divide $1$ by $1$ .

$m^{6} \cdot 1 + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.6

Multiply $m^{6}$ by $1$ .

$m^{6} + 6 {(m)}^{5} {(\frac{1}{m})}^{1} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.7

Simplify.

$m^{6} + 6 m^{5} \frac{1}{m} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.8

Cancel the common factor of $m$ .

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Step 4.8.1

Factor $m$ out of $6 m^{5}$ .

$m^{6} + m (6 m^{4}) \frac{1}{m} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.8.2

Cancel the common factor.

Step 4.8.3

Rewrite the expression.

$m^{6} + 6 m^{4} + 15 {(m)}^{4} {(\frac{1}{m})}^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.9

Apply the product rule to $\frac{1}{m}$ .

$m^{6} + 6 m^{4} + 15 m^{4} \frac{1^{2}}{m^{2}} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.10

One to any power is one.

$m^{6} + 6 m^{4} + 15 m^{4} \frac{1}{m^{2}} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.11

Cancel the common factor of $m^{2}$ .

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Step 4.11.1

Factor $m^{2}$ out of $15 m^{4}$ .

$m^{6} + 6 m^{4} + m^{2} (15 m^{2}) \frac{1}{m^{2}} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.11.2

Cancel the common factor.

$m^{6} + 6 m^{4} + m^{2} (15 m^{2}) \frac{1}{m^{2}} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.11.3

Rewrite the expression.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 {(m)}^{3} {(\frac{1}{m})}^{3} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.12

Apply the product rule to $\frac{1}{m}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 m^{3} \frac{1^{3}}{m^{3}} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.13

One to any power is one.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 m^{3} \frac{1}{m^{3}} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.14

Cancel the common factor of $m^{3}$ .

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Step 4.14.1

Factor $m^{3}$ out of $20 m^{3}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + m^{3} \cdot 20 \frac{1}{m^{3}} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.14.2

Cancel the common factor.

$m^{6} + 6 m^{4} + 15 m^{2} + m^{3} \cdot 20 \frac{1}{m^{3}} + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.14.3

Rewrite the expression.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + 15 {(m)}^{2} {(\frac{1}{m})}^{4} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.15

Apply the product rule to $\frac{1}{m}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + 15 m^{2} \frac{1^{4}}{m^{4}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.16

One to any power is one.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + 15 m^{2} \frac{1}{m^{4}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.17

Cancel the common factor of $m^{2}$ .

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Step 4.17.1

Factor $m^{2}$ out of $15 m^{2}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + m^{2} \cdot 15 \frac{1}{m^{4}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.17.2

Factor $m^{2}$ out of $m^{4}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + m^{2} \cdot 15 \frac{1}{m^{2} m^{2}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.17.3

Cancel the common factor.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + m^{2} \cdot 15 \frac{1}{m^{2} m^{2}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.17.4

Rewrite the expression.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + 15 \frac{1}{m^{2}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.18

Combine $15$ and $\frac{1}{m^{2}}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + 6 {(m)}^{1} {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.19

Simplify.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + 6 m {(\frac{1}{m})}^{5} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.20

Apply the product rule to $\frac{1}{m}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + 6 m \frac{1^{5}}{m^{5}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.21

One to any power is one.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + 6 m \frac{1}{m^{5}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.22

Cancel the common factor of $m$ .

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Step 4.22.1

Factor $m$ out of $6 m$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + m \cdot 6 \frac{1}{m^{5}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.22.2

Factor $m$ out of $m^{5}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + m \cdot 6 \frac{1}{m \cdot m^{4}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.22.3

Cancel the common factor.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + m \cdot 6 \frac{1}{m \cdot m^{4}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.22.4

Rewrite the expression.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + 6 \frac{1}{m^{4}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.23

Combine $6$ and $\frac{1}{m^{4}}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + 1 {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.24

Multiply ${(m)}^{0}$ by $1$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + {(m)}^{0} {(\frac{1}{m})}^{6}$

Step 4.25

Anything raised to $0$ is $1$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + 1 {(\frac{1}{m})}^{6}$

Step 4.26

Multiply ${(\frac{1}{m})}^{6}$ by $1$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + {(\frac{1}{m})}^{6}$

Step 4.27

Apply the product rule to $\frac{1}{m}$ .

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + \frac{1^{6}}{m^{6}}$

Step 4.28

One to any power is one.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + \frac{1}{m^{6}}$