Basic Math Examples

$\frac{- \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}}{35 \sqrt{35 - p^{2}}} = 2$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$2 \cdot (35 \sqrt{35 - p^{2}}) = - \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $2 \cdot (35 \sqrt{35 - p^{2}})$ .

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Step 1.2.1.1

Remove parentheses.

$2 \cdot (35 \sqrt{35 - p^{2}}) = - \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}$

Step 1.2.1.2

Multiply $35$ by $2$ .

$70 \sqrt{35 - p^{2}} = - \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify $- \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}$ .

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Step 1.3.1.1

Multiply $p$ by $1$ .

$70 \sqrt{35 - p^{2}} = - \frac{p}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}$

Step 1.3.1.2

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

$70 \sqrt{35 - p^{2}} = - \frac{p}{2} \cdot \frac{1}{{(32 - p^{2})}^{\frac{1}{2}}}$

Step 1.3.1.3

Multiply $\frac{1}{{(32 - p^{2})}^{\frac{1}{2}}}$ by $\frac{p}{2}$ .

$70 \sqrt{35 - p^{2}} = - \frac{p}{{(32 - p^{2})}^{\frac{1}{2}} \cdot 2}$

Step 1.3.1.4

Move $2$ to the left of ${(32 - p^{2})}^{\frac{1}{2}}$ .

$70 \sqrt{35 - p^{2}} = - \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}}$

Step 2

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

${(70 \sqrt{35 - p^{2}})}^{2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

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 $\sqrt{35 - p^{2}}$ as ${(35 - p^{2})}^{\frac{1}{2}}$ .

${(70 {(35 - p^{2})}^{\frac{1}{2}})}^{2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(70 {(35 - p^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $70 {(35 - p^{2})}^{\frac{1}{2}}$ .

$70^{2} {({(35 - p^{2})}^{\frac{1}{2}})}^{2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.2

Raise $70$ to the power of $2$ .

$4900 {({(35 - p^{2})}^{\frac{1}{2}})}^{2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(35 - p^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

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

$4900 {(35 - p^{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

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.

$4900 {(35 - p^{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$4900 {(35 - p^{2})}^{1} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.4

Simplify.

$4900 (35 - p^{2}) = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.5

Apply the distributive property.

$4900 \cdot 35 + 4900 (- p^{2}) = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.6

Multiply.

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Step 3.2.1.6.1

Multiply $4900$ by $35$ .

$171500 + 4900 (- p^{2}) = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.2.1.6.2

Multiply $- 1$ by $4900$ .

$171500 - 4900 p^{2} = {(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$ .

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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 $- \frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}}$ .

$171500 - 4900 p^{2} = {(- 1)}^{2} {(\frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}})}^{2}$

Step 3.3.1.1.2

Apply the product rule to $\frac{p}{2 {(32 - p^{2})}^{\frac{1}{2}}}$ .

$171500 - 4900 p^{2} = {(- 1)}^{2} \frac{p^{2}}{{(2 {(32 - p^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.1.3

Apply the product rule to $2 {(32 - p^{2})}^{\frac{1}{2}}$ .

$171500 - 4900 p^{2} = {(- 1)}^{2} \frac{p^{2}}{2^{2} {({(32 - p^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.2

Simplify the expression.

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Step 3.3.1.2.1

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

$171500 - 4900 p^{2} = 1 \frac{p^{2}}{2^{2} {({(32 - p^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.2.2

Multiply $\frac{p^{2}}{2^{2} {({(32 - p^{2})}^{\frac{1}{2}})}^{2}}$ by $1$ .

$171500 - 4900 p^{2} = \frac{p^{2}}{2^{2} {({(32 - p^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.3

Simplify the denominator.

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Step 3.3.1.3.1

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

$171500 - 4900 p^{2} = \frac{p^{2}}{4 {({(32 - p^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.3.1.3.2

Multiply the exponents in ${({(32 - p^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.3.2.1

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

$171500 - 4900 p^{2} = \frac{p^{2}}{4 {(32 - p^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.3.1.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.1.3.2.2.1

Cancel the common factor.

$171500 - 4900 p^{2} = \frac{p^{2}}{4 {(32 - p^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.3.1.3.2.2.2

Rewrite the expression.

$171500 - 4900 p^{2} = \frac{p^{2}}{4 {(32 - p^{2})}^{1}}$

Step 3.3.1.3.3

Simplify.

$171500 - 4900 p^{2} = \frac{p^{2}}{4 (32 - p^{2})}$

Step 4

Solve for $p$ .

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Step 4.1

Subtract $171500$ from both sides of the equation.

$- 4900 p^{2} = \frac{p^{2}}{4 (32 - p^{2})} - 171500$

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.

$1, 4 (32 - p^{2}), 1$

Step 4.2.2

The LCM of one and any expression is the expression.

$4 (32 - p^{2})$

Step 4.3

Multiply each term in $- 4900 p^{2} = \frac{p^{2}}{4 (32 - p^{2})} - 171500$ by $4 (32 - p^{2})$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $- 4900 p^{2} = \frac{p^{2}}{4 (32 - p^{2})} - 171500$ by $4 (32 - p^{2})$ .

$- 4900 p^{2} (4 (32 - p^{2})) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

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.

$- 4900 p^{2} (4 \cdot 32 + 4 (- p^{2})) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.1.2

Multiply.

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Step 4.3.2.1.2.1

Multiply $4$ by $32$ .

$- 4900 p^{2} (128 + 4 (- p^{2})) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.1.2.2

Multiply $- 1$ by $4$ .

$- 4900 p^{2} (128 - 4 p^{2}) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.1.3

Apply the distributive property.

$- 4900 p^{2} \cdot 128 - 4900 p^{2} (- 4 p^{2}) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.1.4

Simplify the expression.

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Step 4.3.2.1.4.1

Multiply $128$ by $- 4900$ .

$- 627200 p^{2} - 4900 p^{2} (- 4 p^{2}) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.1.4.2

Rewrite using the commutative property of multiplication.

$- 627200 p^{2} - 4900 \cdot - 4 p^{2} p^{2} = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.2

Simplify each term.

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Step 4.3.2.2.1

Multiply $p^{2}$ by $p^{2}$ by adding the exponents.

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Step 4.3.2.2.1.1

Move $p^{2}$ .

$- 627200 p^{2} - 4900 \cdot - 4 (p^{2} p^{2}) = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.2.1.2

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

$- 627200 p^{2} - 4900 \cdot - 4 p^{2 + 2} = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.2.1.3

Add $2$ and $2$ .

$- 627200 p^{2} - 4900 \cdot - 4 p^{4} = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

Step 4.3.2.2.2

Multiply $- 4900$ by $- 4$ .

$- 627200 p^{2} + 19600 p^{4} = \frac{p^{2}}{4 (32 - p^{2})} (4 (32 - p^{2})) - 171500 (4 (32 - p^{2}))$

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

Rewrite using the commutative property of multiplication.

$- 627200 p^{2} + 19600 p^{4} = 4 \frac{p^{2}}{4 (32 - p^{2})} (32 - p^{2}) - 171500 (4 (32 - p^{2}))$

Step 4.3.3.1.2

Cancel the common factor of $4$ .

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Step 4.3.3.1.2.1

Cancel the common factor.

$- 627200 p^{2} + 19600 p^{4} = 4 \frac{p^{2}}{4 (32 - p^{2})} (32 - p^{2}) - 171500 (4 (32 - p^{2}))$

Step 4.3.3.1.2.2

Rewrite the expression.

$- 627200 p^{2} + 19600 p^{4} = \frac{p^{2}}{32 - p^{2}} (32 - p^{2}) - 171500 (4 (32 - p^{2}))$

Step 4.3.3.1.3

Cancel the common factor of $32 - p^{2}$ .

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Step 4.3.3.1.3.1

Cancel the common factor.

$- 627200 p^{2} + 19600 p^{4} = \frac{p^{2}}{32 - p^{2}} (32 - p^{2}) - 171500 (4 (32 - p^{2}))$

Step 4.3.3.1.3.2

Rewrite the expression.

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 171500 (4 (32 - p^{2}))$

Step 4.3.3.1.4

Apply the distributive property.

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 171500 (4 \cdot 32 + 4 (- p^{2}))$

Step 4.3.3.1.5

Multiply $4$ by $32$ .

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 171500 (128 + 4 (- p^{2}))$

Step 4.3.3.1.6

Multiply $- 1$ by $4$ .

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 171500 (128 - 4 p^{2})$

Step 4.3.3.1.7

Apply the distributive property.

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 171500 \cdot 128 - 171500 (- 4 p^{2})$

Step 4.3.3.1.8

Multiply $- 171500$ by $128$ .

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 21952000 - 171500 (- 4 p^{2})$

Step 4.3.3.1.9

Multiply $- 4$ by $- 171500$ .

$- 627200 p^{2} + 19600 p^{4} = p^{2} - 21952000 + 686000 p^{2}$

Step 4.3.3.2

Add $p^{2}$ and $686000 p^{2}$ .

$- 627200 p^{2} + 19600 p^{4} = 686001 p^{2} - 21952000$

Step 4.4

Solve the equation.

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Step 4.4.1

Move all the expressions to the left side of the equation.

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Step 4.4.1.1

Subtract $686001 p^{2}$ from both sides of the equation.

$- 627200 p^{2} + 19600 p^{4} - 686001 p^{2} = - 21952000$

Step 4.4.1.2

Add $21952000$ to both sides of the equation.

$- 627200 p^{2} + 19600 p^{4} - 686001 p^{2} + 21952000 = 0$

Step 4.4.2

Subtract $686001 p^{2}$ from $- 627200 p^{2}$ .

$19600 p^{4} - 1313201 p^{2} + 21952000 = 0$

Step 4.4.3

Substitute $u = p^{2}$ into the equation. This will make the quadratic formula easy to use.

$19600 u^{2} - 1313201 u + 21952000 = 0$

$u = p^{2}$

Step 4.4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4.5

Substitute the values $a = 19600$ , $b = - 1313201$ , and $c = 21952000$ into the quadratic formula and solve for $u$ .

$\frac{1313201 \pm \sqrt{{(- 1313201)}^{2} - 4 \cdot (19600 \cdot 21952000)}}{2 \cdot 19600}$

Step 4.4.6

Simplify.

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Step 4.4.6.1

Simplify the numerator.

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Step 4.4.6.1.1

Raise $- 1313201$ to the power of $2$ .

$u = \frac{1313201 \pm \sqrt{1724496866401 - 4 \cdot 19600 \cdot 21952000}}{2 \cdot 19600}$

Step 4.4.6.1.2

Multiply $- 4 \cdot 19600 \cdot 21952000$ .

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Step 4.4.6.1.2.1

Multiply $- 4$ by $19600$ .

$u = \frac{1313201 \pm \sqrt{1724496866401 - 78400 \cdot 21952000}}{2 \cdot 19600}$

Step 4.4.6.1.2.2

Multiply $- 78400$ by $21952000$ .

$u = \frac{1313201 \pm \sqrt{1724496866401 - 1721036800000}}{2 \cdot 19600}$

Step 4.4.6.1.3

Subtract $1721036800000$ from $1724496866401$ .

$u = \frac{1313201 \pm \sqrt{3460066401}}{2 \cdot 19600}$

Step 4.4.6.2

Multiply $2$ by $19600$ .

$u = \frac{1313201 \pm \sqrt{3460066401}}{39200}$

Step 4.4.7

The final answer is the combination of both solutions.

$u = \frac{1313201 + \sqrt{3460066401}}{39200}, \frac{1313201 - \sqrt{3460066401}}{39200}$

Step 4.4.8

Substitute the real value of $u = p^{2}$ back into the solved equation.

$p^{2} = 35.00059513$

${(p^{2})}^{1} = 31.99945589$

Step 4.4.9

Solve the first equation for $p$ .

$p^{2} = 35.00059513$

Step 4.4.10

Solve the equation for $p$ .

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Step 4.4.10.1

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

$p = \pm \sqrt{35.00059513}$

Step 4.4.10.2

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

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Step 4.4.10.2.1

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

$p = \sqrt{35.00059513}$

Step 4.4.10.2.2

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

$p = - \sqrt{35.00059513}$

Step 4.4.10.2.3

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

$p = \sqrt{35.00059513}, - \sqrt{35.00059513}$

Step 4.4.11

Solve the second equation for $p$ .

${(p^{2})}^{1} = 31.99945589$

Step 4.4.12

Solve the equation for $p$ .

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Step 4.4.12.1

Remove parentheses.

$p^{2} = 31.99945589$

Step 4.4.12.2

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

$p = \pm \sqrt{31.99945589}$

Step 4.4.12.3

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

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Step 4.4.12.3.1

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

$p = \sqrt{31.99945589}$

Step 4.4.12.3.2

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

$p = - \sqrt{31.99945589}$

Step 4.4.12.3.3

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

$p = \sqrt{31.99945589}, - \sqrt{31.99945589}$

Step 4.4.13

The solution to $19600 p^{4} - 1313201 p^{2} + 21952000 = 0$ is $p = \sqrt{35.00059513}, - \sqrt{35.00059513}, \sqrt{31.99945589}, - \sqrt{31.99945589}$ .

$p = \sqrt{35.00059513}, - \sqrt{35.00059513}, \sqrt{31.99945589}, - \sqrt{31.99945589}$

Step 5

Exclude the solutions that do not make $\frac{- \frac{p \cdot 1}{2} \cdot {(32 - p^{2})}^{- \frac{1}{2}}}{35 \sqrt{35 - p^{2}}} = 2$ true.

$p = - \sqrt{31.99945589}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$p = - \sqrt{31.99945589}$

Decimal Form:

$p = - 5.65680615 \dots$