Basic Math Examples

sup $(a b) = s u p a \cdot (s u p (b))$

Step 1

Rewrite the equation as $s u p a \cdot (s u p b) = a b$ .

$s u p a \cdot (s u p b) = a b$

Step 2

Simplify.

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Step 2.1

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

$u p a \cdot (s^{1} s u p b) = a b$

Step 2.2

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

$u p a \cdot (s^{1} s^{1} u p b) = a b$

Step 2.3

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

$u p a \cdot (s^{1 + 1} u p b) = a b$

Step 2.4

Add $1$ and $1$ .

$u p a \cdot (s^{2} u p b) = a b$

Step 2.5

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

$p a \cdot (s^{2} (u^{1} u) p b) = a b$

Step 2.6

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

$p a \cdot (s^{2} (u^{1} u^{1}) p b) = a b$

Step 2.7

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

$p a \cdot (s^{2} u^{1 + 1} p b) = a b$

Step 2.8

Add $1$ and $1$ .

$p a \cdot (s^{2} u^{2} p b) = a b$

Step 2.9

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

$p^{1} p a \cdot (s^{2} u^{2} b) = a b$

Step 2.10

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

$p^{1} p^{1} a \cdot (s^{2} u^{2} b) = a b$

Step 2.11

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

$p^{1 + 1} a \cdot (s^{2} u^{2} b) = a b$

Step 2.12

Add $1$ and $1$ .

$p^{2} a \cdot (s^{2} u^{2} b) = a b$

Step 2.13

Remove parentheses.

$p^{2} a \cdot (s^{2} u^{2}) b = a b$

Step 2.14

Remove parentheses.

$p^{2} a \cdot s^{2} u^{2} b = a b$

Step 3

Divide each term in $p^{2} a \cdot (s^{2} u^{2} b) = a b$ by $p^{2} a u^{2} b$ and simplify.

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Step 3.1

Divide each term in $p^{2} a \cdot (s^{2} u^{2} b) = a b$ by $p^{2} a u^{2} b$ .

$\frac{p^{2} a \cdot (s^{2} u^{2} b)}{p^{2} a u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $p^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{p^{2} a \cdot (s^{2} u^{2} b)}{p^{2} a u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.1.2

Rewrite the expression.

$\frac{a \cdot (s^{2} u^{2} b)}{a u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.2

Cancel the common factor of $a$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{a \cdot (s^{2} u^{2} b)}{a u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.2.2

Rewrite the expression.

$\frac{s^{2} u^{2} b}{u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.3

Cancel the common factor of $u^{2}$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{s^{2} u^{2} b}{u^{2} b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.3.2

Rewrite the expression.

$\frac{s^{2} b}{b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.4

Cancel the common factor of $b$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{s^{2} b}{b} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.2.4.2

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

$s^{2} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $a$ .

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Step 3.3.1.1

Cancel the common factor.

$s^{2} = \frac{a b}{p^{2} a u^{2} b}$

Step 3.3.1.2

Rewrite the expression.

$s^{2} = \frac{b}{p^{2} u^{2} b}$

Step 3.3.2

Cancel the common factor of $b$ .

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Step 3.3.2.1

Cancel the common factor.

$s^{2} = \frac{b}{p^{2} u^{2} b}$

Step 3.3.2.2

Rewrite the expression.

$s^{2} = \frac{1}{p^{2} u^{2}}$

Step 4

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

$s = \pm \sqrt{\frac{1}{p^{2} u^{2}}}$

Step 5

Simplify $\pm \sqrt{\frac{1}{p^{2} u^{2}}}$ .

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Step 5.1

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

$s = \pm \sqrt{\frac{1^{2}}{p^{2} u^{2}}}$

Step 5.2

Rewrite $p^{2} u^{2}$ as ${(p u)}^{2}$ .

$s = \pm \sqrt{\frac{1^{2}}{{(p u)}^{2}}}$

Step 5.3

Rewrite $\frac{1^{2}}{{(p u)}^{2}}$ as ${(\frac{1}{p u})}^{2}$ .

$s = \pm \sqrt{{(\frac{1}{p u})}^{2}}$

Step 5.4

Pull terms out from under the radical, assuming positive real numbers.

$s = \pm \frac{1}{p u}$

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.

$s = \frac{1}{p u}$

Step 6.2

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

$s = - \frac{1}{p u}$

Step 6.3

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

$s = \frac{1}{p u}$

$s = - \frac{1}{p u}$

$s = \frac{1}{p u}$

$s = - \frac{1}{p u}$