Precalculus Examples

$3^{2 t - 5} = \sqrt{27}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27}$ as $27^{\frac{1}{2}}$ .

$3^{2 t - 5} = 27^{\frac{1}{2}}$

Step 2

Create equivalent expressions in the equation that all have equal bases.

$3^{2 t - 5} = 3^{3 (\frac{1}{2})}$

Step 3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 t - 5 = 3 (\frac{1}{2})$

Step 4

Solve for $t$ .

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Step 4.1

Simplify $3 (\frac{1}{2})$ .

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Step 4.1.1

Rewrite.

$2 t - 5 = 0 + 0 + 3 (\frac{1}{2})$

Step 4.1.2

Simplify by adding zeros.

$2 t - 5 = 3 (\frac{1}{2})$

Step 4.1.3

Combine $3$ and $\frac{1}{2}$ .

$2 t - 5 = \frac{3}{2}$

Step 4.2

Move all terms not containing $t$ to the right side of the equation.

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Step 4.2.1

Add $5$ to both sides of the equation.

$2 t = \frac{3}{2} + 5$

Step 4.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 t = \frac{3}{2} + 5 \cdot \frac{2}{2}$

Step 4.2.3

Combine $5$ and $\frac{2}{2}$ .

$2 t = \frac{3}{2} + \frac{5 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$2 t = \frac{3 + 5 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $5$ by $2$ .

$2 t = \frac{3 + 10}{2}$

Step 4.2.5.2

Add $3$ and $10$ .

$2 t = \frac{13}{2}$

Step 4.3

Divide each term in $2 t = \frac{13}{2}$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 t = \frac{13}{2}$ by $2$ .

$\frac{2 t}{2} = \frac{\frac{13}{2}}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{\frac{13}{2}}{2}$

Step 4.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{\frac{13}{2}}{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{13}{2} \cdot \frac{1}{2}$

Step 4.3.3.2

Multiply $\frac{13}{2} \cdot \frac{1}{2}$ .

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Step 4.3.3.2.1

Multiply $\frac{13}{2}$ by $\frac{1}{2}$ .

$t = \frac{13}{2 \cdot 2}$

Step 4.3.3.2.2

Multiply $2$ by $2$ .

$t = \frac{13}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{13}{4}$

Decimal Form:

$t = 3.25$

Mixed Number Form:

$t = 3 \frac{1}{4}$