Basic Math Examples

$\frac{1}{8} = 16 \cdot 2^{\frac{t}{5730}}$

Step 1

Rewrite the equation as $16 \cdot 2^{\frac{t}{5730}} = \frac{1}{8}$ .

$16 \cdot 2^{\frac{t}{5730}} = \frac{1}{8}$

Step 2

Raise $8$ to the power of $1$ .

$16 \cdot 2^{\frac{t}{5730}} = \frac{1}{8^{1}}$

Step 3

Move $8^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$16 \cdot 2^{\frac{t}{5730}} = 8^{- 1}$

Step 4

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

$2^{4} \cdot 2^{\frac{t}{5730}} = 8^{- 1}$

Step 5

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

$2^{4 + \frac{t}{5730}} = 8^{- 1}$

Step 6

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

$2^{4 + \frac{t}{5730}} = 2^{- 3}$

Step 7

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

$4 + \frac{t}{5730} = - 3$

Step 8

Solve for $t$ .

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Step 8.1

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

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Step 8.1.1

Subtract $4$ from both sides of the equation.

$\frac{t}{5730} = - 3 - 4$

Step 8.1.2

Subtract $4$ from $- 3$ .

$\frac{t}{5730} = - 7$

Step 8.2

Multiply both sides of the equation by $5730$ .

$5730 \frac{t}{5730} = 5730 \cdot - 7$

Step 8.3

Simplify both sides of the equation.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $5730$ .

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Step 8.3.1.1.1

Cancel the common factor.

$5730 \frac{t}{5730} = 5730 \cdot - 7$

Step 8.3.1.1.2

Rewrite the expression.

$t = 5730 \cdot - 7$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Multiply $5730$ by $- 7$ .

$t = - 40110$