Basic Math Examples

$\frac{\sqrt{3}}{2} y - \frac{1}{10} y = 25 \sqrt{3}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{\sqrt{3}}{2}$ and $y$ .

$\frac{\sqrt{3} y}{2} - \frac{1}{10} y = 25 \sqrt{3}$

Step 1.2

Combine $y$ and $\frac{1}{10}$ .

$\frac{\sqrt{3} y}{2} - \frac{y}{10} = 25 \sqrt{3}$

Step 2

Multiply each term in $\frac{\sqrt{3} y}{2} - \frac{y}{10} = 25 \sqrt{3}$ by $10$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{\sqrt{3} y}{2} - \frac{y}{10} = 25 \sqrt{3}$ by $10$ .

$\frac{\sqrt{3} y}{2} \cdot 10 - \frac{y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Factor $2$ out of $10$ .

$\frac{\sqrt{3} y}{2} \cdot (2 (5)) - \frac{y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{\sqrt{3} y}{2} \cdot (2 \cdot 5) - \frac{y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.1.3

Rewrite the expression.

$\sqrt{3} y \cdot 5 - \frac{y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.2

Move $5$ to the left of $\sqrt{3} y$ .

$5 \cdot (\sqrt{3} y) - \frac{y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.3

Cancel the common factor of $10$ .

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Step 2.2.1.3.1

Move the leading negative in $- \frac{y}{10}$ into the numerator.

$5 \sqrt{3} y + \frac{- y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.3.2

Cancel the common factor.

$5 \sqrt{3} y + \frac{- y}{10} \cdot 10 = 25 \sqrt{3} \cdot 10$

Step 2.2.1.3.3

Rewrite the expression.

$5 \sqrt{3} y - y = 25 \sqrt{3} \cdot 10$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $10$ by $25$ .

$5 \sqrt{3} y - y = 250 \sqrt{3}$

Step 3

Factor $y$ out of $5 \sqrt{3} y - y$ .

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Step 3.1

Factor $y$ out of $5 \sqrt{3} y$ .

$y (5 \sqrt{3}) - y = 250 \sqrt{3}$

Step 3.2

Factor $y$ out of $- y$ .

$y (5 \sqrt{3}) + y \cdot - 1 = 250 \sqrt{3}$

Step 3.3

Factor $y$ out of $y (5 \sqrt{3}) + y \cdot - 1$ .

$y (5 \sqrt{3} - 1) = 250 \sqrt{3}$

Step 4

Divide each term in $y (5 \sqrt{3} - 1) = 250 \sqrt{3}$ by $5 \sqrt{3} - 1$ and simplify.

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Step 4.1

Divide each term in $y (5 \sqrt{3} - 1) = 250 \sqrt{3}$ by $5 \sqrt{3} - 1$ .

$\frac{y (5 \sqrt{3} - 1)}{5 \sqrt{3} - 1} = \frac{250 \sqrt{3}}{5 \sqrt{3} - 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5 \sqrt{3} - 1$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{y (5 \sqrt{3} - 1)}{5 \sqrt{3} - 1} = \frac{250 \sqrt{3}}{5 \sqrt{3} - 1}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{250 \sqrt{3}}{5 \sqrt{3} - 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $\frac{250 \sqrt{3}}{5 \sqrt{3} - 1}$ by $\frac{5 \sqrt{3} + 1}{5 \sqrt{3} + 1}$ .

$y = \frac{250 \sqrt{3}}{5 \sqrt{3} - 1} \cdot \frac{5 \sqrt{3} + 1}{5 \sqrt{3} + 1}$

Step 4.3.2

Multiply $\frac{250 \sqrt{3}}{5 \sqrt{3} - 1}$ by $\frac{5 \sqrt{3} + 1}{5 \sqrt{3} + 1}$ .

$y = \frac{250 \sqrt{3} (5 \sqrt{3} + 1)}{(5 \sqrt{3} - 1) (5 \sqrt{3} + 1)}$

Step 4.3.3

Expand the denominator using the FOIL method.

$y = \frac{250 \sqrt{3} (5 \sqrt{3} + 1)}{25 {\sqrt{3}}^{2} + 5 \sqrt{3} - 5 \sqrt{3} - 1}$

Step 4.3.4

Simplify.

$y = \frac{250 \sqrt{3} (5 \sqrt{3} + 1)}{74}$

Step 4.3.5

Cancel the common factor of $250$ and $74$ .

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Step 4.3.5.1

Factor $2$ out of $250 \sqrt{3} (5 \sqrt{3} + 1)$ .

$y = \frac{2 (125 \sqrt{3} (5 \sqrt{3} + 1))}{74}$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $2$ out of $74$ .

$y = \frac{2 (125 \sqrt{3} (5 \sqrt{3} + 1))}{2 (37)}$

Step 4.3.5.2.2

Cancel the common factor.

$y = \frac{2 (125 \sqrt{3} (5 \sqrt{3} + 1))}{2 \cdot 37}$

Step 4.3.5.2.3

Rewrite the expression.

$y = \frac{125 \sqrt{3} (5 \sqrt{3} + 1)}{37}$

Step 4.3.6

Group $5 \sqrt{3} + 1$ and $\sqrt{3}$ together.

$y = \frac{125 ((5 \sqrt{3} + 1) \sqrt{3})}{37}$

Step 4.3.7

Apply the distributive property.

$y = \frac{125 (5 \sqrt{3} \sqrt{3} + 1 \sqrt{3})}{37}$

Step 4.3.8

Multiply $5 \sqrt{3} \sqrt{3}$ .

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Step 4.3.8.1

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

$y = \frac{125 (5 ({\sqrt{3}}^{1} \sqrt{3}) + 1 \sqrt{3})}{37}$

Step 4.3.8.2

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

$y = \frac{125 (5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 1 \sqrt{3})}{37}$

Step 4.3.8.3

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

$y = \frac{125 (5 {\sqrt{3}}^{1 + 1} + 1 \sqrt{3})}{37}$

Step 4.3.8.4

Add $1$ and $1$ .

$y = \frac{125 (5 {\sqrt{3}}^{2} + 1 \sqrt{3})}{37}$

Step 4.3.9

Multiply $\sqrt{3}$ by $1$ .

$y = \frac{125 (5 {\sqrt{3}}^{2} + \sqrt{3})}{37}$

Step 4.3.10

Simplify each term.

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Step 4.3.10.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.10.1.1

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

$y = \frac{125 (5 {(3^{\frac{1}{2}})}^{2} + \sqrt{3})}{37}$

Step 4.3.10.1.2

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

$y = \frac{125 (5 \cdot 3^{\frac{1}{2} \cdot 2} + \sqrt{3})}{37}$

Step 4.3.10.1.3

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

$y = \frac{125 (5 \cdot 3^{\frac{2}{2}} + \sqrt{3})}{37}$

Step 4.3.10.1.4

Cancel the common factor of $2$ .

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Step 4.3.10.1.4.1

Cancel the common factor.

$y = \frac{125 (5 \cdot 3^{\frac{2}{2}} + \sqrt{3})}{37}$

Step 4.3.10.1.4.2

Rewrite the expression.

$y = \frac{125 (5 \cdot 3^{1} + \sqrt{3})}{37}$

Step 4.3.10.1.5

Evaluate the exponent.

$y = \frac{125 (5 \cdot 3 + \sqrt{3})}{37}$

Step 4.3.10.2

Multiply $5$ by $3$ .

$y = \frac{125 (15 + \sqrt{3})}{37}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{125 (15 + \sqrt{3})}{37}$

Decimal Form:

$y = 56.52719867 \dots$