Basic Math Examples

$645 = 3 n^{\frac{3}{2}} - 3$

Step 1

Rewrite the equation as $3 n^{\frac{3}{2}} - 3 = 645$ .

$3 n^{\frac{3}{2}} - 3 = 645$

Step 2

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

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Step 2.1

Add $3$ to both sides of the equation.

$3 n^{\frac{3}{2}} - 645 = 3$

Step 2.2

Add $645$ to both sides of the equation.

$3 n^{\frac{3}{2}} = 3 + 645$

Step 2.3

Add $3$ and $645$ .

$3 n^{\frac{3}{2}} = 648$

Step 3

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(3 n^{\frac{3}{2}})}^{\frac{2}{3}} = 648^{\frac{2}{3}}$

Step 4

Simplify the left side.

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Step 4.1

Simplify ${(3 n^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 4.1.1

Apply the product rule to $3 n^{\frac{3}{2}}$ .

$3^{\frac{2}{3}} {(n^{\frac{3}{2}})}^{\frac{2}{3}} = 648^{\frac{2}{3}}$

Step 4.1.2

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

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Step 4.1.2.1

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

$3^{\frac{2}{3}} n^{\frac{3}{2} \cdot \frac{2}{3}} = 648^{\frac{2}{3}}$

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

$3^{\frac{2}{3}} n^{\frac{3}{2} \cdot \frac{2}{3}} = 648^{\frac{2}{3}}$

Step 4.1.2.2.2

Rewrite the expression.

$3^{\frac{2}{3}} n^{\frac{1}{2} \cdot 2} = 648^{\frac{2}{3}}$

Step 4.1.2.3

Cancel the common factor of $2$ .

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Step 4.1.2.3.1

Cancel the common factor.

$3^{\frac{2}{3}} n^{\frac{1}{2} \cdot 2} = 648^{\frac{2}{3}}$

Step 4.1.2.3.2

Rewrite the expression.

$3^{\frac{2}{3}} n^{1} = 648^{\frac{2}{3}}$

Step 4.1.3

Simplify.

$3^{\frac{2}{3}} n = 648^{\frac{2}{3}}$

Step 4.1.4

Reorder factors in $3^{\frac{2}{3}} n$ .

$n \cdot 3^{\frac{2}{3}} = 648^{\frac{2}{3}}$

Step 5

Divide each term in $n \cdot 3^{\frac{2}{3}} = 648^{\frac{2}{3}}$ by $3^{\frac{2}{3}}$ and simplify.

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Step 5.1

Divide each term in $n \cdot 3^{\frac{2}{3}} = 648^{\frac{2}{3}}$ by $3^{\frac{2}{3}}$ .

$\frac{n \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}} = \frac{648^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor.

$\frac{n \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}} = \frac{648^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Step 5.2.2

Divide $n$ by $1$ .

$n = \frac{648^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$n = {(\frac{648}{3})}^{\frac{2}{3}}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Divide $648$ by $3$ .

$n = 216^{\frac{2}{3}}$

Step 5.3.2.2

Rewrite $216$ as $6^{3}$ .

$n = {(6^{3})}^{\frac{2}{3}}$

Step 5.3.2.3

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

$n = 6^{3 (\frac{2}{3})}$

Step 5.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.1

Cancel the common factor.

$n = 6^{3 (\frac{2}{3})}$

Step 5.3.3.2

Rewrite the expression.

$n = 6^{2}$

Step 5.3.4

Raise $6$ to the power of $2$ .

$n = 36$