Basic Math Examples

$\frac{d}{\frac{3}{5}} \cdot d = - 6$

Step 1

Simplify $\frac{d}{\frac{3}{5}} \cdot d$ .

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$d \frac{5}{3} \cdot d = - 6$

Step 1.2

Multiply $d$ by $d$ by adding the exponents.

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Step 1.2.1

Move $d$ .

$d \cdot d \frac{5}{3} = - 6$

Step 1.2.2

Multiply $d$ by $d$ .

$d^{2} \frac{5}{3} = - 6$

Step 1.3

Combine $d^{2}$ and $\frac{5}{3}$ .

$\frac{d^{2} \cdot 5}{3} = - 6$

Step 1.4

Move $5$ to the left of $d^{2}$ .

$\frac{5 d^{2}}{3} = - 6$

Step 2

Multiply both sides of the equation by $\frac{3}{5}$ .

$\frac{3}{5} \cdot \frac{5 d^{2}}{3} = \frac{3}{5} \cdot - 6$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3}{5} \cdot \frac{5 d^{2}}{3}$ .

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Step 3.1.1.1

Combine.

$\frac{3 (5 d^{2})}{5 \cdot 3} = \frac{3}{5} \cdot - 6$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{3 (5 d^{2})}{5 \cdot 3} = \frac{3}{5} \cdot - 6$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{5 d^{2}}{5} = \frac{3}{5} \cdot - 6$

Step 3.1.1.3

Cancel the common factor of $5$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{5 d^{2}}{5} = \frac{3}{5} \cdot - 6$

Step 3.1.1.3.2

Divide $d^{2}$ by $1$ .

$d^{2} = \frac{3}{5} \cdot - 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{3}{5} \cdot - 6$ .

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Step 3.2.1.1

Multiply $\frac{3}{5} \cdot - 6$ .

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Step 3.2.1.1.1

Combine $\frac{3}{5}$ and $- 6$ .

$d^{2} = \frac{3 \cdot - 6}{5}$

Step 3.2.1.1.2

Multiply $3$ by $- 6$ .

$d^{2} = \frac{- 18}{5}$

Step 3.2.1.2

Move the negative in front of the fraction.

$d^{2} = - \frac{18}{5}$

Step 4

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

$d = \pm \sqrt{- \frac{18}{5}}$

Step 5

Simplify $\pm \sqrt{- \frac{18}{5}}$ .

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Step 5.1

Rewrite $- 1$ as $i^{2}$ .

$d = \pm \sqrt{i^{2} \frac{18}{5}}$

Step 5.2

Pull terms out from under the radical.

$d = \pm i \sqrt{\frac{18}{5}}$

Step 5.3

Rewrite $\sqrt{\frac{18}{5}}$ as $\frac{\sqrt{18}}{\sqrt{5}}$ .

$d = \pm i \frac{\sqrt{18}}{\sqrt{5}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 5.4.1.1

Factor $9$ out of $18$ .

$d = \pm i \frac{\sqrt{9 (2)}}{\sqrt{5}}$

Step 5.4.1.2

Rewrite $9$ as $3^{2}$ .

$d = \pm i \frac{\sqrt{3^{2} \cdot 2}}{\sqrt{5}}$

Step 5.4.2

Pull terms out from under the radical.

$d = \pm i \frac{3 \sqrt{2}}{\sqrt{5}}$

Step 5.5

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$d = \pm i (\frac{3 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.6.2

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

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 5.6.3

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

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 5.6.4

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

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.6.5

Add $1$ and $1$ .

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.6.6.2

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

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.6.6.4.2

Rewrite the expression.

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{5^{1}}$

Step 5.6.6.5

Evaluate the exponent.

$d = \pm i \frac{3 \sqrt{2} \sqrt{5}}{5}$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Combine using the product rule for radicals.

$d = \pm i \frac{3 \sqrt{5 \cdot 2}}{5}$

Step 5.7.2

Multiply $5$ by $2$ .

$d = \pm i \frac{3 \sqrt{10}}{5}$

Step 5.8

Combine fractions.

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Step 5.8.1

Combine $i$ and $\frac{3 \sqrt{10}}{5}$ .

$d = \pm \frac{i (3 \sqrt{10})}{5}$

Step 5.8.2

Move $3$ to the left of $i$ .

$d = \pm \frac{3 i \sqrt{10}}{5}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$d = \frac{3 i \sqrt{10}}{5}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$d = - \frac{3 i \sqrt{10}}{5}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$d = \frac{3 i \sqrt{10}}{5}, - \frac{3 i \sqrt{10}}{5}$