Algebra Examples

${(y z + 1)}^{5}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(y z + 1)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $y z$ and $b$ $1$ into the expression.

$1 {(y z)}^{5} {(1)}^{0} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4

Simplify each term.

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Step 4.1

Multiply $1$ by ${(1)}^{0}$ by adding the exponents.

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Step 4.1.1

Move ${(1)}^{0}$ .

${(1)}^{0} \cdot 1 {(y z)}^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.1.2

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

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Step 4.1.2.1

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

${(1)}^{0} \cdot 1^{1} {(y z)}^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.1.2.2

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

$1^{0 + 1} {(y z)}^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.1.3

Add $0$ and $1$ .

$1^{1} {(y z)}^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.2

Simplify $1^{1} {(y z)}^{5}$ .

${(y z)}^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.3

Apply the product rule to $y z$ .

$y^{5} z^{5} + 5 {(y z)}^{4} {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.4

Apply the product rule to $y z$ .

$y^{5} z^{5} + 5 (y^{4} z^{4}) {(1)}^{1} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.5

Evaluate the exponent.

$y^{5} z^{5} + 5 y^{4} z^{4} \cdot 1 + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.6

Multiply $5$ by $1$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 {(y z)}^{3} {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.7

Apply the product rule to $y z$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 (y^{3} z^{3}) {(1)}^{2} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.8

One to any power is one.

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} \cdot 1 + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.9

Multiply $10$ by $1$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 {(y z)}^{2} {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.10

Apply the product rule to $y z$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 (y^{2} z^{2}) {(1)}^{3} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.11

One to any power is one.

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} \cdot 1 + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.12

Multiply $10$ by $1$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 {(y z)}^{1} {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.13

Simplify.

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 (y z) {(1)}^{4} + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.14

One to any power is one.

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z \cdot 1 + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.15

Multiply $5$ by $1$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + 1 {(y z)}^{0} {(1)}^{5}$

Step 4.16

Multiply $1$ by ${(1)}^{5}$ by adding the exponents.

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Step 4.16.1

Move ${(1)}^{5}$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + {(1)}^{5} \cdot 1 {(y z)}^{0}$

Step 4.16.2

Multiply ${(1)}^{5}$ by $1$ .

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Step 4.16.2.1

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

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + {(1)}^{5} \cdot 1^{1} {(y z)}^{0}$

Step 4.16.2.2

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

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + 1^{5 + 1} {(y z)}^{0}$

Step 4.16.3

Add $5$ and $1$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + 1^{6} {(y z)}^{0}$

Step 4.17

Simplify $1^{6} {(y z)}^{0}$ .

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + 1^{6}$

Step 4.18

One to any power is one.

$y^{5} z^{5} + 5 y^{4} z^{4} + 10 y^{3} z^{3} + 10 y^{2} z^{2} + 5 y z + 1$