Calculus Examples

$\frac{27}{16} = t^{2}$

Step 1

Rewrite the equation as $t^{2} = \frac{27}{16}$ .

$t^{2} = \frac{27}{16}$

Step 2

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

$t = \pm \sqrt{\frac{27}{16}}$

Step 3

Simplify $\pm \sqrt{\frac{27}{16}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{27}{16}}$ as $\frac{\sqrt{27}}{\sqrt{16}}$ .

$t = \pm \frac{\sqrt{27}}{\sqrt{16}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.2.1.1

Factor $9$ out of $27$ .

$t = \pm \frac{\sqrt{9 (3)}}{\sqrt{16}}$

Step 3.2.1.2

Rewrite $9$ as $3^{2}$ .

$t = \pm \frac{\sqrt{3^{2} \cdot 3}}{\sqrt{16}}$

Step 3.2.2

Pull terms out from under the radical.

$t = \pm \frac{3 \sqrt{3}}{\sqrt{16}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $16$ as $4^{2}$ .

$t = \pm \frac{3 \sqrt{3}}{\sqrt{4^{2}}}$

Step 3.3.2

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

$t = \pm \frac{3 \sqrt{3}}{4}$

Step 4

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

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Step 4.1

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

$t = \frac{3 \sqrt{3}}{4}$

Step 4.2

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

$t = - \frac{3 \sqrt{3}}{4}$

Step 4.3

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

$t = \frac{3 \sqrt{3}}{4}, - \frac{3 \sqrt{3}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{3 \sqrt{3}}{4}, - \frac{3 \sqrt{3}}{4}$

Decimal Form:

$t = 1.29903810 \dots, - 1.29903810 \dots$