Finite Math Examples

$1350 = 16 t^{2}$

Step 1

Rewrite the equation as $16 t^{2} = 1350$ .

$16 t^{2} = 1350$

Step 2

Divide each term in $16 t^{2} = 1350$ by $16$ and simplify.

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Step 2.1

Divide each term in $16 t^{2} = 1350$ by $16$ .

$\frac{16 t^{2}}{16} = \frac{1350}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $16$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 t^{2}}{16} = \frac{1350}{16}$

Step 2.2.1.2

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

$t^{2} = \frac{1350}{16}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $1350$ and $16$ .

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Step 2.3.1.1

Factor $2$ out of $1350$ .

$t^{2} = \frac{2 (675)}{16}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $16$ .

$t^{2} = \frac{2 \cdot 675}{2 \cdot 8}$

Step 2.3.1.2.2

Cancel the common factor.

$t^{2} = \frac{2 \cdot 675}{2 \cdot 8}$

Step 2.3.1.2.3

Rewrite the expression.

$t^{2} = \frac{675}{8}$

Step 3

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

$t = \pm \sqrt{\frac{675}{8}}$

Step 4

Simplify $\pm \sqrt{\frac{675}{8}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{675}{8}}$ as $\frac{\sqrt{675}}{\sqrt{8}}$ .

$t = \pm \frac{\sqrt{675}}{\sqrt{8}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $675$ as $15^{2} \cdot 3$ .

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Step 4.2.1.1

Factor $225$ out of $675$ .

$t = \pm \frac{\sqrt{225 (3)}}{\sqrt{8}}$

Step 4.2.1.2

Rewrite $225$ as $15^{2}$ .

$t = \pm \frac{\sqrt{15^{2} \cdot 3}}{\sqrt{8}}$

Step 4.2.2

Pull terms out from under the radical.

$t = \pm \frac{15 \sqrt{3}}{\sqrt{8}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.3.1.1

Factor $4$ out of $8$ .

$t = \pm \frac{15 \sqrt{3}}{\sqrt{4 (2)}}$

Step 4.3.1.2

Rewrite $4$ as $2^{2}$ .

$t = \pm \frac{15 \sqrt{3}}{\sqrt{2^{2} \cdot 2}}$

Step 4.3.2

Pull terms out from under the radical.

$t = \pm \frac{15 \sqrt{3}}{2 \sqrt{2}}$

Step 4.4

Multiply $\frac{15 \sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = \pm \frac{15 \sqrt{3}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{15 \sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 4.5.2

Move $\sqrt{2}$ .

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 4.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 4.5.5

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

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 4.5.6

Add $1$ and $1$ .

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 4.5.7

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

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Step 4.5.7.1

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

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 4.5.7.2

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

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.5.7.3

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

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.5.7.4

Cancel the common factor of $2$ .

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Step 4.5.7.4.1

Cancel the common factor.

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.5.7.4.2

Rewrite the expression.

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \cdot 2^{1}}$

Step 4.5.7.5

Evaluate the exponent.

$t = \pm \frac{15 \sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Combine using the product rule for radicals.

$t = \pm \frac{15 \sqrt{2 \cdot 3}}{2 \cdot 2}$

Step 4.6.2

Multiply $2$ by $3$ .

$t = \pm \frac{15 \sqrt{6}}{2 \cdot 2}$

Step 4.7

Multiply $2$ by $2$ .

$t = \pm \frac{15 \sqrt{6}}{4}$

Step 5

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

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Step 5.1

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

$t = \frac{15 \sqrt{6}}{4}$

Step 5.2

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

$t = - \frac{15 \sqrt{6}}{4}$

Step 5.3

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

$t = \frac{15 \sqrt{6}}{4}, - \frac{15 \sqrt{6}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{15 \sqrt{6}}{4}, - \frac{15 \sqrt{6}}{4}$

Decimal Form:

$t = 9.18558653 \dots, - 9.18558653 \dots$