Basic Math Examples

10.4 = \frac{1}{2} \cdot {9.8}^{t^{2}}

$10.4 = \frac{1}{2} \cdot {9.8}^{t^{2}}$

Step 1

Rewrite the equation as

\frac{1}{2} \cdot {9.8}^{t^{2}} = 10.4

$\frac{1}{2} \cdot {9.8}^{t^{2}} = 10.4$ .

\frac{1}{2} \cdot {9.8}^{t^{2}} = 10.4

$\frac{1}{2} \cdot {9.8}^{t^{2}} = 10.4$

Step 2

Multiply both sides of the equation by

2

$2$ .

2 (\frac{1}{2} \cdot {9.8}^{t^{2}}) = 2 \cdot 10.4

$2 (\frac{1}{2} \cdot {9.8}^{t^{2}}) = 2 \cdot 10.4$

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

2 (\frac{1}{2} \cdot {9.8}^{t^{2}})

$2 (\frac{1}{2} \cdot {9.8}^{t^{2}})$ .

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Step 3.1.1.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

{9.8}^{t^{2}}

${9.8}^{t^{2}}$ .

2 \frac{{9.8}^{t^{2}}}{2} = 2 \cdot 10.4

$2 \frac{{9.8}^{t^{2}}}{2} = 2 \cdot 10.4$

Step 3.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$2 \frac{{9.8}^{t^{2}}}{2} = 2 \cdot 10.4$

Step 3.1.1.2.2

Rewrite the expression.

${9.8}^{t^{2}} = 2 \cdot 10.4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply

$2$ by

$10.4$ .

${9.8}^{t^{2}} = 20.8$

Step 4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({9.8}^{t^{2}}) = \ln (20.8)$

Step 5

Expand

$\ln ({9.8}^{t^{2}})$ by moving

$t^{2}$ outside the logarithm.

$t^{2} \ln (9.8) = \ln (20.8)$

Step 6

Divide each term in

$t^{2} \ln (9.8) = \ln (20.8)$ by

$\ln (9.8)$ and simplify.

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Step 6.1

Divide each term in

$t^{2} \ln (9.8) = \ln (20.8)$ by

$\ln (9.8)$ .

$\frac{t^{2} \ln (9.8)}{\ln (9.8)} = \frac{\ln (20.8)}{\ln (9.8)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

$\ln (9.8)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{t^{2} \ln (9.8)}{\ln (9.8)} = \frac{\ln (20.8)}{\ln (9.8)}$

Step 6.2.1.2

Divide

$t^{2}$ by

$1$ .

$t^{2} = \frac{\ln (20.8)}{\ln (9.8)}$

Step 7

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

$t = \pm \sqrt{\frac{\ln (20.8)}{\ln (9.8)}}$

Step 8

Simplify

$\pm \sqrt{\frac{\ln (20.8)}{\ln (9.8)}}$ .

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Step 8.1

Rewrite

$\sqrt{\frac{\ln (20.8)}{\ln (9.8)}}$ as

$\frac{\sqrt{\ln (20.8)}}{\sqrt{\ln (9.8)}}$ .

$t = \pm \frac{\sqrt{\ln (20.8)}}{\sqrt{\ln (9.8)}}$

Step 8.2

Multiply

$\frac{\sqrt{\ln (20.8)}}{\sqrt{\ln (9.8)}}$ by

$\frac{\sqrt{\ln (9.8)}}{\sqrt{\ln (9.8)}}$ .

$t = \pm \frac{\sqrt{\ln (20.8)}}{\sqrt{\ln (9.8)}} \cdot \frac{\sqrt{\ln (9.8)}}{\sqrt{\ln (9.8)}}$

Step 8.3

Combine and simplify the denominator.

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Step 8.3.1

Multiply

$\frac{\sqrt{\ln (20.8)}}{\sqrt{\ln (9.8)}}$ by

$\frac{\sqrt{\ln (9.8)}}{\sqrt{\ln (9.8)}}$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{\sqrt{\ln (9.8)} \sqrt{\ln (9.8)}}$

Step 8.3.2

Raise

$\sqrt{\ln (9.8)}$ to the power of

$1$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\sqrt{\ln (9.8)}}^{1} \sqrt{\ln (9.8)}}$

Step 8.3.3

Raise

$\sqrt{\ln (9.8)}$ to the power of

$1$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\sqrt{\ln (9.8)}}^{1} {\sqrt{\ln (9.8)}}^{1}}$

Step 8.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\sqrt{\ln (9.8)}}^{1 + 1}}$

Step 8.3.5

Add

$1$ and

$1$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\sqrt{\ln (9.8)}}^{2}}$

Step 8.3.6

Rewrite

${\sqrt{\ln (9.8)}}^{2}$ as

$\ln (9.8)$ .

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Step 8.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{\ln (9.8)}$ as

${\ln (9.8)}^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{({\ln (9.8)}^{\frac{1}{2}})}^{2}}$

Step 8.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\ln (9.8)}^{\frac{1}{2} \cdot 2}}$

Step 8.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\ln (9.8)}^{\frac{2}{2}}}$

Step 8.3.6.4

Cancel the common factor of

$2$ .

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Step 8.3.6.4.1

Cancel the common factor.

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{{\ln (9.8)}^{\frac{2}{2}}}$

Step 8.3.6.4.2

Rewrite the expression.

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{\ln^{1} (9.8)}$

Step 8.3.6.5

Simplify.

$t = \pm \frac{\sqrt{\ln (20.8)} \sqrt{\ln (9.8)}}{\ln (9.8)}$

Step 8.4

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}$

Step 9

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

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Step 9.1

First, use the positive value of the

$\pm$ to find the first solution.

$t = \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}$

Step 9.2

Next, use the negative value of the

$\pm$ to find the second solution.

$t = - \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}$

Step 9.3

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

$t = \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}, - \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}, - \frac{\sqrt{\ln (20.8) \ln (9.8)}}{\ln (9.8)}$

Decimal Form:

$t = 1.15313931 \dots, - 1.15313931 \dots$