Basic Math Examples

$3 z^{5} - 5 = 12$

Step 1

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

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Step 1.1

Add $5$ to both sides of the equation.

$3 z^{5} = 12 + 5$

Step 1.2

Add $12$ and $5$ .

$3 z^{5} = 17$

Step 2

Divide each term in $3 z^{5} = 17$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 z^{5} = 17$ by $3$ .

$\frac{3 z^{5}}{3} = \frac{17}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 z^{5}}{3} = \frac{17}{3}$

Step 2.2.1.2

Divide $z^{5}$ by $1$ .

$z^{5} = \frac{17}{3}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \sqrt[5]{\frac{17}{3}}$

Step 4

Simplify $\sqrt[5]{\frac{17}{3}}$ .

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Step 4.1

Rewrite $\sqrt[5]{\frac{17}{3}}$ as $\frac{\sqrt[5]{17}}{\sqrt[5]{3}}$ .

$z = \frac{\sqrt[5]{17}}{\sqrt[5]{3}}$

Step 4.2

Multiply $\frac{\sqrt[5]{17}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$z = \frac{\sqrt[5]{17}}{\sqrt[5]{3}} \cdot \frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{\sqrt[5]{17}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{\sqrt[5]{3} {\sqrt[5]{3}}^{4}}$

Step 4.3.2

Raise $\sqrt[5]{3}$ to the power of $1$ .

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1} {\sqrt[5]{3}}^{4}}$

Step 4.3.3

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

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1 + 4}}$

Step 4.3.4

Add $1$ and $4$ .

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{5}}$

Step 4.3.5

Rewrite ${\sqrt[5]{3}}^{5}$ as $3$ .

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Step 4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3}$ as $3^{\frac{1}{5}}$ .

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{{(3^{\frac{1}{5}})}^{5}}$

Step 4.3.5.2

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

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{3^{\frac{1}{5} \cdot 5}}$

Step 4.3.5.3

Combine $\frac{1}{5}$ and $5$ .

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 4.3.5.4

Cancel the common factor of $5$ .

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Step 4.3.5.4.1

Cancel the common factor.

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 4.3.5.4.2

Rewrite the expression.

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{3^{1}}$

Step 4.3.5.5

Evaluate the exponent.

$z = \frac{\sqrt[5]{17} {\sqrt[5]{3}}^{4}}{3}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Rewrite ${\sqrt[5]{3}}^{4}$ as $\sqrt[5]{3^{4}}$ .

$z = \frac{\sqrt[5]{17} \sqrt[5]{3^{4}}}{3}$

Step 4.4.2

Raise $3$ to the power of $4$ .

$z = \frac{\sqrt[5]{17} \sqrt[5]{81}}{3}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Combine using the product rule for radicals.

$z = \frac{\sqrt[5]{17 \cdot 81}}{3}$

Step 4.5.2

Multiply $17$ by $81$ .

$z = \frac{\sqrt[5]{1377}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$z = \frac{\sqrt[5]{1377}}{3}$

Decimal Form:

$z = 1.41470384 \dots$