Basic Math Examples

$\sqrt{15 - z} + 6 = \sqrt{4 - 9}$

Step 1

Solve for $\sqrt{15 - z}$ .

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Step 1.1

Simplify $\sqrt{4 - 9}$ .

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Step 1.1.1

Rewrite.

$\sqrt{15 - z} + 6 = 0 + 0 + \sqrt{4 - 9}$

Step 1.1.2

Subtract $9$ from $4$ .

$\sqrt{15 - z} + 6 = \sqrt{- 5}$

Step 1.1.3

Rewrite $- 5$ as $- 1 (5)$ .

$\sqrt{15 - z} + 6 = \sqrt{- 1 (5)}$

Step 1.1.4

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$\sqrt{15 - z} + 6 = \sqrt{- 1} \cdot \sqrt{5}$

Step 1.1.5

Rewrite $\sqrt{- 1}$ as $i$ .

$\sqrt{15 - z} + 6 = i \sqrt{5}$

Step 1.2

Move all terms not containing $\sqrt{15 - z}$ to the right side of the equation.

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Step 1.2.1

Subtract $6$ from both sides of the equation.

$\sqrt{15 - z} = i \sqrt{5} - 6$

Step 1.2.2

Reorder $i \sqrt{5}$ and $- 6$ .

$\sqrt{15 - z} = - 6 + i \sqrt{5}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{15 - z}}^{2} = {(- 6 + i \sqrt{5})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 - z}$ as ${(15 - z)}^{\frac{1}{2}}$ .

${({(15 - z)}^{\frac{1}{2}})}^{2} = {(- 6 + i \sqrt{5})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(15 - z)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(15 - z)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(15 - z)}^{\frac{1}{2} \cdot 2} = {(- 6 + i \sqrt{5})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(15 - z)}^{\frac{1}{2} \cdot 2} = {(- 6 + i \sqrt{5})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(15 - z)}^{1} = {(- 6 + i \sqrt{5})}^{2}$

Step 3.2.1.2

Simplify.

$15 - z = {(- 6 + i \sqrt{5})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 6 + i \sqrt{5})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- 6 + i \sqrt{5})}^{2}$ as $(- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$ .

$15 - z = (- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$

Step 3.3.1.2

Expand $(- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$15 - z = - 6 (- 6 + i \sqrt{5}) + i \sqrt{5} (- 6 + i \sqrt{5})$

Step 3.3.1.2.2

Apply the distributive property.

$15 - z = - 6 \cdot - 6 - 6 (i \sqrt{5}) + i \sqrt{5} (- 6 + i \sqrt{5})$

Step 3.3.1.2.3

Apply the distributive property.

$15 - z = - 6 \cdot - 6 - 6 (i \sqrt{5}) + i \sqrt{5} \cdot - 6 + i \sqrt{5} (i \sqrt{5})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $- 6$ by $- 6$ .

$15 - z = 36 - 6 (i \sqrt{5}) + i \sqrt{5} \cdot - 6 + i \sqrt{5} (i \sqrt{5})$

Step 3.3.1.3.1.2

Move $- 6$ to the left of $i \sqrt{5}$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 \cdot (i \sqrt{5}) + i \sqrt{5} (i \sqrt{5})$

Step 3.3.1.3.1.3

Multiply $i \sqrt{5} (i \sqrt{5})$ .

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Step 3.3.1.3.1.3.1

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1} i \sqrt{5} \sqrt{5}$

Step 3.3.1.3.1.3.2

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1} i^{1} \sqrt{5} \sqrt{5}$

Step 3.3.1.3.1.3.3

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1 + 1} \sqrt{5} \sqrt{5}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} \sqrt{5} \sqrt{5}$

Step 3.3.1.3.1.3.5

Raise $\sqrt{5}$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

Step 3.3.1.3.1.3.6

Raise $\sqrt{5}$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 3.3.1.3.1.3.7

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} {\sqrt{5}}^{1 + 1}$

Step 3.3.1.3.1.3.8

Add $1$ and $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} {\sqrt{5}}^{2}$

Step 3.3.1.3.1.4

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 {\sqrt{5}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 {(5^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{1}$

Step 3.3.1.3.1.5.5

Evaluate the exponent.

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5$

Step 3.3.1.3.1.6

Multiply $- 1$ by $5$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 5$

Step 3.3.1.3.2

Subtract $5$ from $36$ .

$15 - z = - 6 i \sqrt{5} - 6 i \sqrt{5} + 31$

Step 3.3.1.3.3

Subtract $6 i \sqrt{5}$ from $- 6 i \sqrt{5}$ .

$15 - z = - 12 i \sqrt{5} + 31$

Step 3.3.1.4

Reorder $- 12 i \sqrt{5}$ and $31$ .

$15 - z = 31 - 12 i \sqrt{5}$

Step 4

Solve for $z$ .

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Step 4.1

Move all terms not containing $z$ to the right side of the equation.

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Step 4.1.1

Subtract $15$ from both sides of the equation.

$- z = 31 - 12 i \sqrt{5} - 15$

Step 4.1.2

Subtract $15$ from $31$ .

$- z = - 12 i \sqrt{5} + 16$

Step 4.1.3

Reorder $- 12 i \sqrt{5}$ and $16$ .

$- z = 16 - 12 i \sqrt{5}$

Step 4.2

Divide each term in $- z = 16 - 12 i \sqrt{5}$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- z = 16 - 12 i \sqrt{5}$ by $- 1$ .

$\frac{- z}{- 1} = \frac{16}{- 1} + \frac{- 12 i \sqrt{5}}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} = \frac{16}{- 1} + \frac{- 12 i \sqrt{5}}{- 1}$

Step 4.2.2.2

Divide $z$ by $1$ .

$z = \frac{16}{- 1} + \frac{- 12 i \sqrt{5}}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Divide $16$ by $- 1$ .

$z = - 16 + \frac{- 12 i \sqrt{5}}{- 1}$

Step 4.2.3.1.2

Move the negative one from the denominator of $\frac{- 12 i \sqrt{5}}{- 1}$ .

$z = - 16 - 1 \cdot (- 12 i \sqrt{5})$

Step 4.2.3.1.3

Rewrite $- 1 \cdot (- 12 i \sqrt{5})$ as $- (- 12 i \sqrt{5})$ .

$z = - 16 - (- 12 i \sqrt{5})$

Step 4.2.3.1.4

Multiply $- 12$ by $- 1$ .

$z = - 16 + 12 i \sqrt{5}$