Basic Math Examples

${(z - 10)}^{\frac{2}{5}} = {(9 z)}^{\frac{1}{5}}$

Step 1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${({(z - 10)}^{\frac{2}{5}})}^{5} = {({(9 z)}^{\frac{1}{5}})}^{5}$

Step 2

Multiply the exponents in ${({(z - 10)}^{\frac{2}{5}})}^{5}$ .

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Step 2.1

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

${(z - 10)}^{\frac{2}{5} \cdot 5} = {({(9 z)}^{\frac{1}{5}})}^{5}$

Step 2.2

Cancel the common factor of $5$ .

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Step 2.2.1

Cancel the common factor.

${(z - 10)}^{\frac{2}{5} \cdot 5} = {({(9 z)}^{\frac{1}{5}})}^{5}$

Step 2.2.2

Rewrite the expression.

${(z - 10)}^{2} = {({(9 z)}^{\frac{1}{5}})}^{5}$

Step 3

Simplify ${({(9 z)}^{\frac{1}{5}})}^{5}$ .

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Step 3.1

Multiply the exponents in ${({(9 z)}^{\frac{1}{5}})}^{5}$ .

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Step 3.1.1

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

${(z - 10)}^{2} = {(9 z)}^{\frac{1}{5} \cdot 5}$

Step 3.1.2

Cancel the common factor of $5$ .

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Step 3.1.2.1

Cancel the common factor.

${(z - 10)}^{2} = {(9 z)}^{\frac{1}{5} \cdot 5}$

Step 3.1.2.2

Rewrite the expression.

${(z - 10)}^{2} = {(9 z)}^{1}$

Step 3.2

Simplify.

${(z - 10)}^{2} = 9 z$

Step 4

Move all terms containing $z$ to the left side of the equation.

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Step 4.1

Subtract $9 z$ from both sides of the equation.

${(z - 10)}^{2} - 9 z = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite ${(z - 10)}^{2}$ as $(z - 10) (z - 10)$ .

$(z - 10) (z - 10) - 9 z = 0$

Step 4.2.2

Expand $(z - 10) (z - 10)$ using the FOIL Method.

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Step 4.2.2.1

Apply the distributive property.

$z (z - 10) - 10 (z - 10) - 9 z = 0$

Step 4.2.2.2

Apply the distributive property.

$z \cdot z + z \cdot - 10 - 10 (z - 10) - 9 z = 0$

Step 4.2.2.3

Apply the distributive property.

$z \cdot z + z \cdot - 10 - 10 z - 10 \cdot - 10 - 9 z = 0$

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $z$ by $z$ .

$z^{2} + z \cdot - 10 - 10 z - 10 \cdot - 10 - 9 z = 0$

Step 4.2.3.1.2

Move $- 10$ to the left of $z$ .

$z^{2} - 10 \cdot z - 10 z - 10 \cdot - 10 - 9 z = 0$

Step 4.2.3.1.3

Multiply $- 10$ by $- 10$ .

$z^{2} - 10 z - 10 z + 100 - 9 z = 0$

Step 4.2.3.2

Subtract $10 z$ from $- 10 z$ .

$z^{2} - 20 z + 100 - 9 z = 0$

Step 4.3

Subtract $9 z$ from $- 20 z$ .

$z^{2} - 29 z + 100 = 0$

Step 5

Factor $z^{2} - 29 z + 100$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $100$ and whose sum is $- 29$ .

$- 25, - 4$

Step 5.2

Write the factored form using these integers.

$(z - 25) (z - 4) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z - 25 = 0$

$z - 4 = 0$

Step 7

Set $z - 25$ equal to $0$ and solve for $z$ .

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Step 7.1

Set $z - 25$ equal to $0$ .

$z - 25 = 0$

Step 7.2

Add $25$ to both sides of the equation.

$z = 25$

Step 8

Set $z - 4$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $z - 4$ equal to $0$ .

$z - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$z = 4$

Step 9

The final solution is all the values that make $(z - 25) (z - 4) = 0$ true.

$z = 25, 4$