Basic Math Examples

$12 z^{\frac{2}{5}} + 7 z^{\frac{1}{5}} = - 1$

Step 1

Find a common factor $z^{\frac{1}{5}}$ that is present in each term.

$12 {(z^{\frac{1}{5}})}^{2} + 7 z^{\frac{1}{5}} + 1$

Step 2

Substitute $u$ for $z^{\frac{1}{5}}$ .

$12 {(u)}^{2} + 7 (u) + 1 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Multiply $7$ by $u$ .

$12 u^{2} + 7 u + 1 = 0$

Step 3.2

Factor by grouping.

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Step 3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot 1 = 12$ and whose sum is $b = 7$ .

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Step 3.2.1.1

Factor $7$ out of $7 u$ .

$12 u^{2} + 7 (u) + 1 = 0$

Step 3.2.1.2

Rewrite $7$ as $3$ plus $4$

$12 u^{2} + (3 + 4) u + 1 = 0$

Step 3.2.1.3

Apply the distributive property.

$12 u^{2} + 3 u + 4 u + 1 = 0$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$(12 u^{2} + 3 u) + 4 u + 1 = 0$

Step 3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$3 u (4 u + 1) + 1 (4 u + 1) = 0$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $4 u + 1$ .

$(4 u + 1) (3 u + 1) = 0$

Step 3.3

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

$4 u + 1 = 0$

$3 u + 1 = 0$

Step 3.4

Set $4 u + 1$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $4 u + 1$ equal to $0$ .

$4 u + 1 = 0$

Step 3.4.2

Solve $4 u + 1 = 0$ for $u$ .

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Step 3.4.2.1

Subtract $1$ from both sides of the equation.

$4 u = - 1$

Step 3.4.2.2

Divide each term in $4 u = - 1$ by $4$ and simplify.

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Step 3.4.2.2.1

Divide each term in $4 u = - 1$ by $4$ .

$\frac{4 u}{4} = \frac{- 1}{4}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{- 1}{4}$

Step 3.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{4}$

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{4}$

Step 3.5

Set $3 u + 1$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $3 u + 1$ equal to $0$ .

$3 u + 1 = 0$

Step 3.5.2

Solve $3 u + 1 = 0$ for $u$ .

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Step 3.5.2.1

Subtract $1$ from both sides of the equation.

$3 u = - 1$

Step 3.5.2.2

Divide each term in $3 u = - 1$ by $3$ and simplify.

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Step 3.5.2.2.1

Divide each term in $3 u = - 1$ by $3$ .

$\frac{3 u}{3} = \frac{- 1}{3}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{- 1}{3}$

Step 3.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{3}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{3}$

Step 3.6

The final solution is all the values that make $(4 u + 1) (3 u + 1) = 0$ true.

$u = - \frac{1}{4}, - \frac{1}{3}$

Step 4

Substitute $z$ for $u$ .

$z^{\frac{1}{5}} = - \frac{1}{4}, - \frac{1}{3}$

Step 5

Solve for $z^{\frac{1}{5}} = - \frac{1}{4}$ for $z$ .

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Step 5.1

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(z^{\frac{1}{5}})}^{5} = {(- \frac{1}{4})}^{5}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

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

$z^{\frac{1}{5} \cdot 5} = {(- \frac{1}{4})}^{5}$

Step 5.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$z^{\frac{1}{5} \cdot 5} = {(- \frac{1}{4})}^{5}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$z^{1} = {(- \frac{1}{4})}^{5}$

Step 5.2.1.1.2

Simplify.

$z = {(- \frac{1}{4})}^{5}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify ${(- \frac{1}{4})}^{5}$ .

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Step 5.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.2.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$z = {(- 1)}^{5} {(\frac{1}{4})}^{5}$

Step 5.2.2.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$z = {(- 1)}^{5} \frac{1^{5}}{4^{5}}$

Step 5.2.2.1.2

Raise $- 1$ to the power of $5$ .

$z = - \frac{1^{5}}{4^{5}}$

Step 5.2.2.1.3

One to any power is one.

$z = - \frac{1}{4^{5}}$

Step 5.2.2.1.4

Raise $4$ to the power of $5$ .

$z = - \frac{1}{1024}$

Step 6

Solve for $z^{\frac{1}{5}} = - \frac{1}{3}$ for $z$ .

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Step 6.1

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(z^{\frac{1}{5}})}^{5} = {(- \frac{1}{3})}^{5}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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Step 6.2.1.1.1.1

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

$z^{\frac{1}{5} \cdot 5} = {(- \frac{1}{3})}^{5}$

Step 6.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$z^{\frac{1}{5} \cdot 5} = {(- \frac{1}{3})}^{5}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$z^{1} = {(- \frac{1}{3})}^{5}$

Step 6.2.1.1.2

Simplify.

$z = {(- \frac{1}{3})}^{5}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${(- \frac{1}{3})}^{5}$ .

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Step 6.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.2.2.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

$z = {(- 1)}^{5} {(\frac{1}{3})}^{5}$

Step 6.2.2.1.1.2

Apply the product rule to $\frac{1}{3}$ .

$z = {(- 1)}^{5} \frac{1^{5}}{3^{5}}$

Step 6.2.2.1.2

Raise $- 1$ to the power of $5$ .

$z = - \frac{1^{5}}{3^{5}}$

Step 6.2.2.1.3

One to any power is one.

$z = - \frac{1}{3^{5}}$

Step 6.2.2.1.4

Raise $3$ to the power of $5$ .

$z = - \frac{1}{243}$

Step 7

List all of the solutions.

$z = - \frac{1}{1024}, - \frac{1}{243}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$z = - \frac{1}{1024}, - \frac{1}{243}$

Decimal Form:

$z = - 0.00097656 \dots, - 0.00411522 \dots$