Algebra Examples

$3 \cdot 5^{2 t - 1} = 75$

Step 1

Subtract $75$ from both sides of the equation.

$3 \cdot 5^{2 t - 1} - 75 = 0$

Step 2

Factor $3$ out of $3 \cdot 5^{2 t - 1} - 75$ .

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Step 2.1

Factor $3$ out of $3 \cdot 5^{2 t - 1}$ .

$3 (5^{2 t - 1}) - 75 = 0$

Step 2.2

Factor $3$ out of $- 75$ .

$3 \cdot 5^{2 t - 1} + 3 \cdot - 25 = 0$

Step 2.3

Factor $3$ out of $3 \cdot 5^{2 t - 1} + 3 \cdot - 25$ .

$3 (5^{2 t - 1} - 25) = 0$

Step 3

Divide each term in $3 (5^{2 t - 1} - 25) = 0$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 (5^{2 t - 1} - 25) = 0$ by $3$ .

$\frac{3 (5^{2 t - 1} - 25)}{3} = \frac{0}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 (5^{2 t - 1} - 25)}{3} = \frac{0}{3}$

Step 3.2.1.2

Divide $5^{2 t - 1} - 25$ by $1$ .

$5^{2 t - 1} - 25 = \frac{0}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $3$ .

$5^{2 t - 1} - 25 = 0$

Step 4

Add $25$ to both sides of the equation.

$5^{2 t - 1} = 25$

Step 5

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

$5^{2 t - 1} = 5^{2}$

Step 6

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

$2 t - 1 = 2$

Step 7

Solve for $t$ .

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Step 7.1

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

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Step 7.1.1

Add $1$ to both sides of the equation.

$2 t = 2 + 1$

Step 7.1.2

Add $2$ and $1$ .

$2 t = 3$

Step 7.2

Divide each term in $2 t = 3$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 t = 3$ by $2$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $t$ by $1$ .

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