Basic Math Examples

$z = c t g \sqrt{a y^{3}}$

Step 1

Rewrite the equation as $c t g \sqrt{a y^{3}} = z$ .

$c t g \sqrt{a y^{3}} = z$

Step 2

Divide each term in $c t g \sqrt{a y^{3}} = z$ by $c t g$ and simplify.

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Step 2.1

Divide each term in $c t g \sqrt{a y^{3}} = z$ by $c t g$ .

$\frac{c t g \sqrt{a y^{3}}}{c t g} = \frac{z}{c t g}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $c$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{c t g \sqrt{a y^{3}}}{c t g} = \frac{z}{c t g}$

Step 2.2.1.2

Rewrite the expression.

$\frac{t g \sqrt{a y^{3}}}{t g} = \frac{z}{c t g}$

Step 2.2.2

Cancel the common factor of $t$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{t g \sqrt{a y^{3}}}{t g} = \frac{z}{c t g}$

Step 2.2.2.2

Rewrite the expression.

$\frac{g \sqrt{a y^{3}}}{g} = \frac{z}{c t g}$

Step 2.2.3

Cancel the common factor of $g$ .

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Step 2.2.3.1

Cancel the common factor.

$\frac{g \sqrt{a y^{3}}}{g} = \frac{z}{c t g}$

Step 2.2.3.2

Divide $\sqrt{a y^{3}}$ by $1$ .

$\sqrt{a y^{3}} = \frac{z}{c t g}$

Step 2.2.4

Rewrite $a y^{3}$ as $y^{2} (a y)$ .

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Step 2.2.4.1

Factor out $y^{2}$ .

$\sqrt{a (y^{2} y)} = \frac{z}{c t g}$

Step 2.2.4.2

Reorder $a$ and $y^{2}$ .

$\sqrt{y^{2} a y} = \frac{z}{c t g}$

Step 2.2.4.3

Add parentheses.

$\sqrt{y^{2} (a y)} = \frac{z}{c t g}$

Step 2.2.5

Pull terms out from under the radical.

$y \sqrt{a y} = \frac{z}{c t g}$

Step 3

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

${(y \sqrt{a y})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${(y {(a y)}^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(y {(a y)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $a y$ .

${(y (a^{\frac{1}{2}} y^{\frac{1}{2}}))}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2

Multiply $y$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1.2.1

Move $y^{\frac{1}{2}}$ .

${(y^{\frac{1}{2}} y a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2.2

Multiply $y^{\frac{1}{2}}$ by $y$ .

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Step 4.2.1.2.2.1

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

${(y^{\frac{1}{2}} y^{1} a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2.2.2

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

${(y^{\frac{1}{2} + 1} a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2.3

Write $1$ as a fraction with a common denominator.

${(y^{\frac{1}{2} + \frac{2}{2}} a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2.4

Combine the numerators over the common denominator.

${(y^{\frac{1 + 2}{2}} a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.2.5

Add $1$ and $2$ .

${(y^{\frac{3}{2}} a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.3

Apply the product rule to $y^{\frac{3}{2}} a^{\frac{1}{2}}$ .

${(y^{\frac{3}{2}})}^{2} {(a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.4

Multiply the exponents in ${(y^{\frac{3}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

$y^{\frac{3}{2} \cdot 2} {(a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

$y^{\frac{3}{2} \cdot 2} {(a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

$y^{3} {(a^{\frac{1}{2}})}^{2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.5

Multiply the exponents in ${(a^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.5.1

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

$y^{3} a^{\frac{1}{2} \cdot 2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.5.2

Cancel the common factor of $2$ .

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Step 4.2.1.5.2.1

Cancel the common factor.

$y^{3} a^{\frac{1}{2} \cdot 2} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.5.2.2

Rewrite the expression.

$y^{3} a^{1} = {(\frac{z}{c t g})}^{2}$

Step 4.2.1.6

Simplify.

$y^{3} a = {(\frac{z}{c t g})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the product rule to $\frac{z}{c t g}$ .

$y^{3} a = \frac{z^{2}}{{(c t g)}^{2}}$

Step 4.3.1.2

Apply the product rule to $c t g$ .

$y^{3} a = \frac{z^{2}}{{(c t)}^{2} g^{2}}$

Step 4.3.1.3

Apply the product rule to $c t$ .

$y^{3} a = \frac{z^{2}}{c^{2} t^{2} g^{2}}$

Step 5

Solve for $y$ .

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Step 5.1

Divide each term in $y^{3} a = \frac{z^{2}}{c^{2} t^{2} g^{2}}$ by $a$ and simplify.

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Step 5.1.1

Divide each term in $y^{3} a = \frac{z^{2}}{c^{2} t^{2} g^{2}}$ by $a$ .

$\frac{y^{3} a}{a} = \frac{\frac{z^{2}}{c^{2} t^{2} g^{2}}}{a}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $a$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{y^{3} a}{a} = \frac{\frac{z^{2}}{c^{2} t^{2} g^{2}}}{a}$

Step 5.1.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{\frac{z^{2}}{c^{2} t^{2} g^{2}}}{a}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$y^{3} = \frac{z^{2}}{c^{2} t^{2} g^{2}} \cdot \frac{1}{a}$

Step 5.1.3.2

Combine.

$y^{3} = \frac{z^{2} \cdot 1}{c^{2} t^{2} g^{2} a}$

Step 5.1.3.3

Multiply $z^{2}$ by $1$ .

$y^{3} = \frac{z^{2}}{c^{2} t^{2} g^{2} a}$

Step 5.2

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

$y = \sqrt[3]{\frac{z^{2}}{c^{2} t^{2} g^{2} a}}$

Step 5.3

Simplify $\sqrt[3]{\frac{z^{2}}{c^{2} t^{2} g^{2} a}}$ .

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Step 5.3.1

Rewrite $\sqrt[3]{\frac{z^{2}}{c^{2} t^{2} g^{2} a}}$ as $\frac{\sqrt[3]{z^{2}}}{\sqrt[3]{c^{2} t^{2} g^{2} a}}$ .

$y = \frac{\sqrt[3]{z^{2}}}{\sqrt[3]{c^{2} t^{2} g^{2} a}}$

Step 5.3.2

Multiply $\frac{\sqrt[3]{z^{2}}}{\sqrt[3]{c^{2} t^{2} g^{2} a}}$ by $\frac{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}$ .

$y = \frac{\sqrt[3]{z^{2}}}{\sqrt[3]{c^{2} t^{2} g^{2} a}} \cdot \frac{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply $\frac{\sqrt[3]{z^{2}}}{\sqrt[3]{c^{2} t^{2} g^{2} a}}$ by $\frac{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}$ .

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{\sqrt[3]{c^{2} t^{2} g^{2} a} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}$

Step 5.3.3.2

Raise $\sqrt[3]{c^{2} t^{2} g^{2} a}$ to the power of $1$ .

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{1} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}$

Step 5.3.3.3

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

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{1 + 2}}$

Step 5.3.3.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{\sqrt[3]{c^{2} t^{2} g^{2} a}}^{3}}$

Step 5.3.3.5

Rewrite ${\sqrt[3]{c^{2} t^{2} g^{2} a}}^{3}$ as $c^{2} t^{2} g^{2} a$ .

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Step 5.3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{c^{2} t^{2} g^{2} a}$ as ${(c^{2} t^{2} g^{2} a)}^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{({(c^{2} t^{2} g^{2} a)}^{\frac{1}{3}})}^{3}}$

Step 5.3.3.5.2

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

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{(c^{2} t^{2} g^{2} a)}^{\frac{1}{3} \cdot 3}}$

Step 5.3.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{(c^{2} t^{2} g^{2} a)}^{\frac{3}{3}}}$

Step 5.3.3.5.4

Cancel the common factor of $3$ .

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Step 5.3.3.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{(c^{2} t^{2} g^{2} a)}^{\frac{3}{3}}}$

Step 5.3.3.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{{(c^{2} t^{2} g^{2} a)}^{1}}$

Step 5.3.3.5.5

Simplify.

$y = \frac{\sqrt[3]{z^{2}} {\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Rewrite ${\sqrt[3]{c^{2} t^{2} g^{2} a}}^{2}$ as $\sqrt[3]{{(c^{2} t^{2} g^{2} a)}^{2}}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c^{2} t^{2} g^{2} a)}^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.2

Apply the product rule to $c^{2} t^{2} g^{2} a$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c^{2} t^{2} g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.3

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

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Step 5.3.4.3.1

Apply the product rule to $c^{2} t^{2} g^{2}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c^{2} t^{2})}^{2} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.3.2

Apply the product rule to $c^{2} t^{2}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c^{2})}^{2} {(t^{2})}^{2} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.4

Multiply the exponents in ${(c^{2})}^{2}$ .

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Step 5.3.4.4.1

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

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{2 \cdot 2} {(t^{2})}^{2} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.4.2

Multiply $2$ by $2$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{4} {(t^{2})}^{2} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.5

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 5.3.4.5.1

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

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{4} t^{2 \cdot 2} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.5.2

Multiply $2$ by $2$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{4} t^{4} {(g^{2})}^{2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.6

Multiply the exponents in ${(g^{2})}^{2}$ .

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Step 5.3.4.6.1

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

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{4} t^{4} g^{2 \cdot 2} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.6.2

Multiply $2$ by $2$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{4} t^{4} g^{4} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7

Rewrite $c^{4} t^{4} g^{4} a^{2}$ as ${(c t g)}^{3} (c t g a^{2})$ .

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Step 5.3.4.7.1

Factor out $c^{3}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{3} c t^{4} g^{4} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.2

Factor out $t^{3}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{3} c (t^{3} t) g^{4} a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.3

Factor out $g^{3}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{3} c (t^{3} t) (g^{3} g) a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.4

Move $t$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{c^{3} c (t^{3}) g^{3} t g a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.5

Move $c$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{(c^{3} (t^{3})) g^{3} c t g a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.6

Rewrite $(c^{3} (t^{3})) g^{3}$ as ${(c t g)}^{3}$ .

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c t g)}^{3} c t g a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.7

Add parentheses.

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c t g)}^{3} c t (g a^{2})}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.8

Add parentheses.

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c t g)}^{3} c (t (g a^{2}))}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.7.9

Add parentheses.

$y = \frac{\sqrt[3]{z^{2}} \sqrt[3]{{(c t g)}^{3} (c t g a^{2})}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.8

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{z^{2}} c t g \sqrt[3]{c t g a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.4.9

Combine using the product rule for radicals.

$y = \frac{c t g \sqrt[3]{z^{2} c t g a^{2}}}{c^{2} t^{2} g^{2} a}$

Step 5.3.5

Reduce the expression by cancelling the common factors.

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Step 5.3.5.1

Cancel the common factor of $c$ and $c^{2}$ .

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Step 5.3.5.1.1

Factor $c$ out of $c t g \sqrt[3]{z^{2} c t g a^{2}}$ .

$y = \frac{c (t g \sqrt[3]{z^{2} c t g a^{2}})}{c^{2} t^{2} g^{2} a}$

Step 5.3.5.1.2

Cancel the common factors.

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Step 5.3.5.1.2.1

Factor $c$ out of $c^{2} t^{2} g^{2} a$ .

$y = \frac{c (t g \sqrt[3]{z^{2} c t g a^{2}})}{c (c t^{2} g^{2} a)}$

Step 5.3.5.1.2.2

Cancel the common factor.

$y = \frac{c (t g \sqrt[3]{z^{2} c t g a^{2}})}{c (c t^{2} g^{2} a)}$

Step 5.3.5.1.2.3

Rewrite the expression.

$y = \frac{t g \sqrt[3]{z^{2} c t g a^{2}}}{c t^{2} g^{2} a}$

Step 5.3.5.2

Cancel the common factor of $t$ and $t^{2}$ .

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Step 5.3.5.2.1

Factor $t$ out of $t g \sqrt[3]{z^{2} c t g a^{2}}$ .

$y = \frac{t (g \sqrt[3]{z^{2} c t g a^{2}})}{c t^{2} g^{2} a}$

Step 5.3.5.2.2

Cancel the common factors.

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Step 5.3.5.2.2.1

Factor $t$ out of $c t^{2} g^{2} a$ .

$y = \frac{t (g \sqrt[3]{z^{2} c t g a^{2}})}{t (c t g^{2} a)}$

Step 5.3.5.2.2.2

Cancel the common factor.

$y = \frac{t (g \sqrt[3]{z^{2} c t g a^{2}})}{t (c t g^{2} a)}$

Step 5.3.5.2.2.3

Rewrite the expression.

$y = \frac{g \sqrt[3]{z^{2} c t g a^{2}}}{c t g^{2} a}$

Step 5.3.5.3

Cancel the common factor of $g$ and $g^{2}$ .

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Step 5.3.5.3.1

Factor $g$ out of $g \sqrt[3]{z^{2} c t g a^{2}}$ .

$y = \frac{g (\sqrt[3]{z^{2} c t g a^{2}})}{c t g^{2} a}$

Step 5.3.5.3.2

Cancel the common factors.

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Step 5.3.5.3.2.1

Factor $g$ out of $c t g^{2} a$ .

$y = \frac{g (\sqrt[3]{z^{2} c t g a^{2}})}{g (c t g a)}$

Step 5.3.5.3.2.2

Cancel the common factor.

$y = \frac{g \sqrt[3]{z^{2} c t g a^{2}}}{g (c t g a)}$

Step 5.3.5.3.2.3

Rewrite the expression.

$y = \frac{\sqrt[3]{z^{2} c t g a^{2}}}{c t g a}$