Basic Math Examples

$z \cdot z = (2 + 3 l) (5 - 2 i)$

Step 1

Rewrite the equation as $(2 + 3 l) (5 - 2 i) = z \cdot z$ .

$(2 + 3 l) (5 - 2 i) = z \cdot z$

Step 2

Simplify.

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Step 2.1

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

$(2 + 3 l) (5 - 2 i) = z^{1} \cdot z$

Step 2.2

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

$(2 + 3 l) (5 - 2 i) = z^{1} \cdot z^{1}$

Step 2.3

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

$(2 + 3 l) (5 - 2 i) = z^{1 + 1}$

Step 2.4

Add $1$ and $1$ .

$(2 + 3 l) (5 - 2 i) = z^{2}$

Step 3

Divide each term in $(2 + 3 l) (5 - 2 i) = z^{2}$ by $5 - 2 i$ and simplify.

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Step 3.1

Divide each term in $(2 + 3 l) (5 - 2 i) = z^{2}$ by $5 - 2 i$ .

$\frac{(2 + 3 l) (5 - 2 i)}{5 - 2 i} = \frac{z^{2}}{5 - 2 i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5 - 2 i$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(2 + 3 l) (5 - 2 i)}{5 - 2 i} = \frac{z^{2}}{5 - 2 i}$

Step 3.2.1.2

Divide $2 + 3 l$ by $1$ .

$2 + 3 l = \frac{z^{2}}{5 - 2 i}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator and denominator of $\frac{z^{2}}{5 - 2 i}$ by the conjugate of $5 - 2 i$ to make the denominator real.

$2 + 3 l = \frac{z^{2}}{5 - 2 i} \cdot \frac{5 + 2 i}{5 + 2 i}$

Step 3.3.2

Multiply.

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Step 3.3.2.1

Combine.

$2 + 3 l = \frac{z^{2} (5 + 2 i)}{(5 - 2 i) (5 + 2 i)}$

Step 3.3.2.2

Simplify the numerator.

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Step 3.3.2.2.1

Apply the distributive property.

$2 + 3 l = \frac{z^{2} \cdot 5 + z^{2} (2 i)}{(5 - 2 i) (5 + 2 i)}$

Step 3.3.2.2.2

Move $5$ to the left of $z^{2}$ .

$2 + 3 l = \frac{5 \cdot z^{2} + z^{2} (2 i)}{(5 - 2 i) (5 + 2 i)}$

Step 3.3.2.2.3

Move $2$ to the left of $z^{2}$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{(5 - 2 i) (5 + 2 i)}$

Step 3.3.2.3

Simplify the denominator.

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Step 3.3.2.3.1

Expand $(5 - 2 i) (5 + 2 i)$ using the FOIL Method.

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Step 3.3.2.3.1.1

Apply the distributive property.

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{5 (5 + 2 i) - 2 i (5 + 2 i)}$

Step 3.3.2.3.1.2

Apply the distributive property.

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{5 \cdot 5 + 5 (2 i) - 2 i (5 + 2 i)}$

Step 3.3.2.3.1.3

Apply the distributive property.

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{5 \cdot 5 + 5 (2 i) - 2 i \cdot 5 - 2 i (2 i)}$

Step 3.3.2.3.2

Simplify.

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Step 3.3.2.3.2.1

Multiply $5$ by $5$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 5 (2 i) - 2 i \cdot 5 - 2 i (2 i)}$

Step 3.3.2.3.2.2

Multiply $2$ by $5$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 2 i \cdot 5 - 2 i (2 i)}$

Step 3.3.2.3.2.3

Multiply $5$ by $- 2$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 2 i (2 i)}$

Step 3.3.2.3.2.4

Multiply $2$ by $- 2$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 4 i i}$

Step 3.3.2.3.2.5

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

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 4 (i^{1} i)}$

Step 3.3.2.3.2.6

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

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 4 (i^{1} i^{1})}$

Step 3.3.2.3.2.7

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

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 4 i^{1 + 1}}$

Step 3.3.2.3.2.8

Add $1$ and $1$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 10 i - 10 i - 4 i^{2}}$

Step 3.3.2.3.2.9

Subtract $10 i$ from $10 i$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 0 - 4 i^{2}}$

Step 3.3.2.3.2.10

Add $25$ and $0$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 - 4 i^{2}}$

Step 3.3.2.3.3

Simplify each term.

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Step 3.3.2.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 - 4 \cdot - 1}$

Step 3.3.2.3.3.2

Multiply $- 4$ by $- 1$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{25 + 4}$

Step 3.3.2.3.4

Add $25$ and $4$ .

$2 + 3 l = \frac{5 z^{2} + 2 z^{2} i}{29}$

Step 3.3.3

Factor $z^{2}$ out of $5 z^{2} + 2 z^{2} i$ .

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Step 3.3.3.1

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

$2 + 3 l = \frac{z^{2} \cdot 5 + 2 z^{2} i}{29}$

Step 3.3.3.2

Factor $z^{2}$ out of $2 z^{2} i$ .

$2 + 3 l = \frac{z^{2} \cdot 5 + z^{2} (2 i)}{29}$

Step 3.3.3.3

Factor $z^{2}$ out of $z^{2} \cdot 5 + z^{2} (2 i)$ .

$2 + 3 l = \frac{z^{2} (5 + 2 i)}{29}$

Step 4

Subtract $2$ from both sides of the equation.

$3 l = \frac{z^{2} (5 + 2 i)}{29} - 2$

Step 5

Divide each term in $3 l = \frac{z^{2} (5 + 2 i)}{29} - 2$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 l = \frac{z^{2} (5 + 2 i)}{29} - 2$ by $3$ .

$\frac{3 l}{3} = \frac{\frac{z^{2} (5 + 2 i)}{29}}{3} + \frac{- 2}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 l}{3} = \frac{\frac{z^{2} (5 + 2 i)}{29}}{3} + \frac{- 2}{3}$

Step 5.2.1.2

Divide $l$ by $1$ .

$l = \frac{\frac{z^{2} (5 + 2 i)}{29}}{3} + \frac{- 2}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$l = \frac{\frac{z^{2} (5 + 2 i)}{29} - 2}{3}$

Step 5.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{29}{29}$ .

$l = \frac{\frac{z^{2} (5 + 2 i)}{29} - 2 \cdot \frac{29}{29}}{3}$

Step 5.3.3

Simplify terms.

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Step 5.3.3.1

Combine $- 2$ and $\frac{29}{29}$ .

$l = \frac{\frac{z^{2} (5 + 2 i)}{29} + \frac{- 2 \cdot 29}{29}}{3}$

Step 5.3.3.2

Combine the numerators over the common denominator.

$l = \frac{\frac{z^{2} (5 + 2 i) - 2 \cdot 29}{29}}{3}$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Apply the distributive property.

$l = \frac{\frac{z^{2} \cdot 5 + z^{2} (2 i) - 2 \cdot 29}{29}}{3}$

Step 5.3.4.2

Move $5$ to the left of $z^{2}$ .

$l = \frac{\frac{5 \cdot z^{2} + z^{2} (2 i) - 2 \cdot 29}{29}}{3}$

Step 5.3.4.3

Move $2$ to the left of $z^{2}$ .

$l = \frac{\frac{5 z^{2} + 2 z^{2} i - 2 \cdot 29}{29}}{3}$

Step 5.3.4.4

Multiply $- 2$ by $29$ .

$l = \frac{\frac{5 z^{2} + 2 z^{2} i - 58}{29}}{3}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$l = \frac{5 z^{2} + 2 z^{2} i - 58}{29} \cdot \frac{1}{3}$

Step 5.3.6

Multiply $\frac{5 z^{2} + 2 z^{2} i - 58}{29} \cdot \frac{1}{3}$ .

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Step 5.3.6.1

Multiply $\frac{5 z^{2} + 2 z^{2} i - 58}{29}$ by $\frac{1}{3}$ .

$l = \frac{5 z^{2} + 2 z^{2} i - 58}{29 \cdot 3}$

Step 5.3.6.2

Multiply $29$ by $3$ .

$l = \frac{5 z^{2} + 2 z^{2} i - 58}{87}$