Basic Math Examples

z = 4 \sqrt{z - 4}

$z = 4 \sqrt{z - 4}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

4 \sqrt{z - 4} = z

$4 \sqrt{z - 4} = z$

Step 2

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

{(4 \sqrt{z - 4})}^{2} = z^{2}

${(4 \sqrt{z - 4})}^{2} = z^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{z - 4}

$\sqrt{z - 4}$ as

{(z - 4)}^{\frac{1}{2}}

${(z - 4)}^{\frac{1}{2}}$ .

{(4 {(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}

${(4 {(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

{(4 {(z - 4)}^{\frac{1}{2}})}^{2}

${(4 {(z - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to

4 {(z - 4)}^{\frac{1}{2}}

$4 {(z - 4)}^{\frac{1}{2}}$ .

4^{2} {({(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}

$4^{2} {({(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}$

Step 3.2.1.2

Raise

4

$4$ to the power of

2

$2$ .

16 {({(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}

$16 {({(z - 4)}^{\frac{1}{2}})}^{2} = z^{2}$

Step 3.2.1.3

Multiply the exponents in

{({(z - 4)}^{\frac{1}{2}})}^{2}

${({(z - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

16 {(z - 4)}^{\frac{1}{2} \cdot 2} = z^{2}

$16 {(z - 4)}^{\frac{1}{2} \cdot 2} = z^{2}$

Step 3.2.1.3.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$16 {(z - 4)}^{\frac{1}{2} \cdot 2} = z^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$16 {(z - 4)}^{1} = z^{2}$

Step 3.2.1.4

Simplify.

$16 (z - 4) = z^{2}$

Step 3.2.1.5

Apply the distributive property.

$16 z + 16 \cdot - 4 = z^{2}$

Step 3.2.1.6

Multiply

$16$ by

$- 4$ .

$16 z - 64 = z^{2}$

Step 4

Solve for

$z$ .

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Step 4.1

Subtract

$z^{2}$ from both sides of the equation.

$16 z - 64 - z^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor

$- 1$ out of

$16 z - 64 - z^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move

$- 64$ .

$16 z - z^{2} - 64 = 0$

Step 4.2.1.1.2

Reorder

$16 z$ and

$- z^{2}$ .

$- z^{2} + 16 z - 64 = 0$

Step 4.2.1.2

Factor

$- 1$ out of

$- z^{2}$ .

$- (z^{2}) + 16 z - 64 = 0$

Step 4.2.1.3

Factor

$- 1$ out of

$16 z$ .

$- (z^{2}) - (- 16 z) - 64 = 0$

Step 4.2.1.4

Rewrite

$- 64$ as

$- 1 (64)$ .

$- (z^{2}) - (- 16 z) - 1 \cdot 64 = 0$

Step 4.2.1.5

Factor

$- 1$ out of

$- (z^{2}) - (- 16 z)$ .

$- (z^{2} - 16 z) - 1 \cdot 64 = 0$

Step 4.2.1.6

Factor

$- 1$ out of

$- (z^{2} - 16 z) - 1 (64)$ .

$- (z^{2} - 16 z + 64) = 0$

Step 4.2.2

Factor using the perfect square rule.

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Step 4.2.2.1

Rewrite

$64$ as

$8^{2}$ .

$- (z^{2} - 16 z + 8^{2}) = 0$

Step 4.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 z = 2 \cdot z \cdot 8$

Step 4.2.2.3

Rewrite the polynomial.

$- (z^{2} - 2 \cdot z \cdot 8 + 8^{2}) = 0$

Step 4.2.2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = z$ and

$b = 8$ .

$- {(z - 8)}^{2} = 0$

Step 4.3

Divide each term in

$- {(z - 8)}^{2} = 0$ by

$- 1$ and simplify.

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Step 4.3.1

Divide each term in

$- {(z - 8)}^{2} = 0$ by

$- 1$ .

$\frac{- {(z - 8)}^{2}}{- 1} = \frac{0}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(z - 8)}^{2}}{1} = \frac{0}{- 1}$

Step 4.3.2.2

Divide

${(z - 8)}^{2}$ by

$1$ .

${(z - 8)}^{2} = \frac{0}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide

$0$ by

$- 1$ .

${(z - 8)}^{2} = 0$

Step 4.4

Set the

$z - 8$ equal to

$0$ .

$z - 8 = 0$

Step 4.5

Add

$8$ to both sides of the equation.

$z = 8$