Basic Math Examples

$5 s {(4 - s)}^{- \frac{1}{2}} - 6 \sqrt{4 - s} = 0$

Step 1

Solve for $- 6 \sqrt{4 - s}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 s \frac{1}{{(4 - s)}^{\frac{1}{2}}} - 6 \sqrt{4 - s} = 0$

Step 1.1.2

Multiply $5 s \frac{1}{{(4 - s)}^{\frac{1}{2}}}$ .

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Step 1.1.2.1

Combine $5$ and $\frac{1}{{(4 - s)}^{\frac{1}{2}}}$ .

$s \frac{5}{{(4 - s)}^{\frac{1}{2}}} - 6 \sqrt{4 - s} = 0$

Step 1.1.2.2

Combine $s$ and $\frac{5}{{(4 - s)}^{\frac{1}{2}}}$ .

$\frac{s \cdot 5}{{(4 - s)}^{\frac{1}{2}}} - 6 \sqrt{4 - s} = 0$

Step 1.1.3

Move $5$ to the left of $s$ .

$\frac{5 s}{{(4 - s)}^{\frac{1}{2}}} - 6 \sqrt{4 - s} = 0$

Step 1.2

Subtract $\frac{5 s}{{(4 - s)}^{\frac{1}{2}}}$ from both sides of the equation.

$- 6 \sqrt{4 - s} = - \frac{5 s}{{(4 - s)}^{\frac{1}{2}}}$

Step 2

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

${(- 6 \sqrt{4 - s})}^{2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - s}$ as ${(4 - s)}^{\frac{1}{2}}$ .

${(- 6 {(4 - s)}^{\frac{1}{2}})}^{2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- 6 {(4 - s)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- 6 {(4 - s)}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(4 - s)}^{\frac{1}{2}})}^{2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.2

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

$36 {({(4 - s)}^{\frac{1}{2}})}^{2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(4 - s)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

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

$36 {(4 - s)}^{\frac{1}{2} \cdot 2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$36 {(4 - s)}^{\frac{1}{2} \cdot 2} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$36 {(4 - s)}^{1} = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.4

Simplify.

$36 (4 - s) = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.5

Apply the distributive property.

$36 \cdot 4 + 36 (- s) = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.6

Multiply.

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Step 3.2.1.6.1

Multiply $36$ by $4$ .

$144 + 36 (- s) = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.6.2

Multiply $- 1$ by $36$ .

$144 - 36 s = {(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$ .

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Apply the product rule to $- \frac{5 s}{{(4 - s)}^{\frac{1}{2}}}$ .

$144 - 36 s = {(- 1)}^{2} {(\frac{5 s}{{(4 - s)}^{\frac{1}{2}}})}^{2}$

Step 3.3.1.1.2

Apply the product rule to $\frac{5 s}{{(4 - s)}^{\frac{1}{2}}}$ .

$144 - 36 s = {(- 1)}^{2} \frac{{(5 s)}^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.1.3

Apply the product rule to $5 s$ .

$144 - 36 s = {(- 1)}^{2} \frac{5^{2} s^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.2

Simplify the expression.

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Step 3.3.1.2.1

Raise $- 1$ to the power of $2$ .

$144 - 36 s = 1 \frac{5^{2} s^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.2.2

Multiply $\frac{5^{2} s^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$ by $1$ .

$144 - 36 s = \frac{5^{2} s^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.2.3

Raise $5$ to the power of $2$ .

$144 - 36 s = \frac{25 s^{2}}{{({(4 - s)}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.3

Simplify the denominator.

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Step 3.3.1.3.1

Multiply the exponents in ${({(4 - s)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.3.1.1

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

$144 - 36 s = \frac{25 s^{2}}{{(4 - s)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.1.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.1

Cancel the common factor.

$144 - 36 s = \frac{25 s^{2}}{{(4 - s)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.1.3.1.2.2

Rewrite the expression.

$144 - 36 s = \frac{25 s^{2}}{{(4 - s)}^{1}}$

Step 3.3.1.3.2

Simplify.

$144 - 36 s = \frac{25 s^{2}}{4 - s}$

Step 4

Solve for $s$ .

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Step 4.1

Subtract $144$ from both sides of the equation.

$- 36 s = \frac{25 s^{2}}{4 - s} - 144$

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 4 - s, 1$

Step 4.2.2

Remove parentheses.

$1, 4 - s, 1$

Step 4.2.3

The LCM of one and any expression is the expression.

$4 - s$

Step 4.3

Multiply each term in $- 36 s = \frac{25 s^{2}}{4 - s} - 144$ by $4 - s$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $- 36 s = \frac{25 s^{2}}{4 - s} - 144$ by $4 - s$ .

$- 36 s (4 - s) = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify by multiplying through.

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Step 4.3.2.1.1

Apply the distributive property.

$- 36 s \cdot 4 - 36 s (- s) = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2.1.2

Simplify the expression.

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Step 4.3.2.1.2.1

Multiply $4$ by $- 36$ .

$- 144 s - 36 s (- s) = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2.1.2.2

Rewrite using the commutative property of multiplication.

$- 144 s - 36 \cdot - 1 s \cdot s = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2.2

Simplify each term.

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Step 4.3.2.2.1

Multiply $s$ by $s$ by adding the exponents.

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Step 4.3.2.2.1.1

Move $s$ .

$- 144 s - 36 \cdot - 1 (s \cdot s) = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2.2.1.2

Multiply $s$ by $s$ .

$- 144 s - 36 \cdot - 1 s^{2} = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.2.2.2

Multiply $- 36$ by $- 1$ .

$- 144 s + 36 s^{2} = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Cancel the common factor of $4 - s$ .

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Step 4.3.3.1.1.1

Cancel the common factor.

$- 144 s + 36 s^{2} = \frac{25 s^{2}}{4 - s} (4 - s) - 144 (4 - s)$

Step 4.3.3.1.1.2

Rewrite the expression.

$- 144 s + 36 s^{2} = 25 s^{2} - 144 (4 - s)$

Step 4.3.3.1.2

Apply the distributive property.

$- 144 s + 36 s^{2} = 25 s^{2} - 144 \cdot 4 - 144 (- s)$

Step 4.3.3.1.3

Multiply $- 144$ by $4$ .

$- 144 s + 36 s^{2} = 25 s^{2} - 576 - 144 (- s)$

Step 4.3.3.1.4

Multiply $- 1$ by $- 144$ .

$- 144 s + 36 s^{2} = 25 s^{2} - 576 + 144 s$

Step 4.4

Solve the equation.

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Step 4.4.1

Move all terms containing $s$ to the left side of the equation.

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Step 4.4.1.1

Subtract $25 s^{2}$ from both sides of the equation.

$- 144 s + 36 s^{2} - 25 s^{2} = - 576 + 144 s$

Step 4.4.1.2

Subtract $144 s$ from both sides of the equation.

$- 144 s + 36 s^{2} - 25 s^{2} - 144 s = - 576$

Step 4.4.1.3

Subtract $144 s$ from $- 144 s$ .

$36 s^{2} - 25 s^{2} - 288 s = - 576$

Step 4.4.1.4

Subtract $25 s^{2}$ from $36 s^{2}$ .

$11 s^{2} - 288 s = - 576$

Step 4.4.2

Add $576$ to both sides of the equation.

$11 s^{2} - 288 s + 576 = 0$

Step 4.4.3

Factor by grouping.

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Step 4.4.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 11 \cdot 576 = 6336$ and whose sum is $b = - 288$ .

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Step 4.4.3.1.1

Factor $- 288$ out of $- 288 s$ .

$11 s^{2} - 288 s + 576 = 0$

Step 4.4.3.1.2

Rewrite $- 288$ as $- 24$ plus $- 264$

$11 s^{2} + (- 24 - 264) s + 576 = 0$

Step 4.4.3.1.3

Apply the distributive property.

$11 s^{2} - 24 s - 264 s + 576 = 0$

Step 4.4.3.2

Factor out the greatest common factor from each group.

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Step 4.4.3.2.1

Group the first two terms and the last two terms.

$(11 s^{2} - 24 s) - 264 s + 576 = 0$

Step 4.4.3.2.2

Factor out the greatest common factor (GCF) from each group.

$s (11 s - 24) - 24 (11 s - 24) = 0$

Step 4.4.3.3

Factor the polynomial by factoring out the greatest common factor, $11 s - 24$ .

$(11 s - 24) (s - 24) = 0$

Step 4.4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$11 s - 24 = 0$

$s - 24 = 0$

Step 4.4.5

Set $11 s - 24$ equal to $0$ and solve for $s$ .

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Step 4.4.5.1

Set $11 s - 24$ equal to $0$ .

$11 s - 24 = 0$

Step 4.4.5.2

Solve $11 s - 24 = 0$ for $s$ .

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Step 4.4.5.2.1

Add $24$ to both sides of the equation.

$11 s = 24$

Step 4.4.5.2.2

Divide each term in $11 s = 24$ by $11$ and simplify.

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Step 4.4.5.2.2.1

Divide each term in $11 s = 24$ by $11$ .

$\frac{11 s}{11} = \frac{24}{11}$

Step 4.4.5.2.2.2

Simplify the left side.

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Step 4.4.5.2.2.2.1

Cancel the common factor of $11$ .

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Step 4.4.5.2.2.2.1.1

Cancel the common factor.

$\frac{11 s}{11} = \frac{24}{11}$

Step 4.4.5.2.2.2.1.2

Divide $s$ by $1$ .

$s = \frac{24}{11}$

Step 4.4.6

Set $s - 24$ equal to $0$ and solve for $s$ .

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Step 4.4.6.1

Set $s - 24$ equal to $0$ .

$s - 24 = 0$

Step 4.4.6.2

Add $24$ to both sides of the equation.

$s = 24$

Step 4.4.7

The final solution is all the values that make $(11 s - 24) (s - 24) = 0$ true.

$s = \frac{24}{11}, 24$

Step 5

Exclude the solutions that do not make $5 s {(4 - s)}^{- \frac{1}{2}} - 6 \sqrt{4 - s} = 0$ true.

$s = \frac{24}{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$s = \frac{24}{11}$

Decimal Form:

$s = 2.\overline{18}$

Mixed Number Form:

$s = 2 \frac{2}{11}$