Basic Math Examples

$122 = 200 {(4)}^{- 0.2 t} + 22$

Step 1

Rewrite the equation as $200 \cdot 4^{- 0.2 t} + 22 = 122$ .

$200 \cdot 4^{- 0.2 t} + 22 = 122$

Step 2

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

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Step 2.1

Subtract $22$ from both sides of the equation.

$200 \cdot 4^{- 0.2 t} = 122 - 22$

Step 2.2

Subtract $22$ from $122$ .

$200 \cdot 4^{- 0.2 t} = 100$

Step 3

Divide each term in $200 \cdot 4^{- 0.2 t} = 100$ by $200$ and simplify.

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Step 3.1

Divide each term in $200 \cdot 4^{- 0.2 t} = 100$ by $200$ .

$\frac{200 \cdot 4^{- 0.2 t}}{200} = \frac{100}{200}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $200$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{200 \cdot 4^{- 0.2 t}}{200} = \frac{100}{200}$

Step 3.2.1.2

Divide $4^{- 0.2 t}$ by $1$ .

$4^{- 0.2 t} = \frac{100}{200}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $100$ and $200$ .

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Step 3.3.1.1

Factor $100$ out of $100$ .

$4^{- 0.2 t} = \frac{100 (1)}{200}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $100$ out of $200$ .

$4^{- 0.2 t} = \frac{100 \cdot 1}{100 \cdot 2}$

Step 3.3.1.2.2

Cancel the common factor.

$4^{- 0.2 t} = \frac{100 \cdot 1}{100 \cdot 2}$

Step 3.3.1.2.3

Rewrite the expression.

$4^{- 0.2 t} = \frac{1}{2}$

Step 4

Raise $2$ to the power of $1$ .

$4^{- 0.2 t} = \frac{1}{2^{1}}$

Step 5

Move $2^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{- 0.2 t} = 2^{- 1}$

Step 6

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

$2^{2 (- 0.2 t)} = 2^{- 1}$

Step 7

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

$2 (- 0.2 t) = - 1$

Step 8

Solve for $t$ .

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Step 8.1

Multiply $- 0.2$ by $2$ .

$- 0.4 t = - 1$

Step 8.2

Divide each term in $- 0.4 t = - 1$ by $- 0.4$ and simplify.

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Step 8.2.1

Divide each term in $- 0.4 t = - 1$ by $- 0.4$ .

$\frac{- 0.4 t}{- 0.4} = \frac{- 1}{- 0.4}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $- 0.4$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{- 0.4 t}{- 0.4} = \frac{- 1}{- 0.4}$

Step 8.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 1}{- 0.4}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $- 1$ by $- 0.4$ .

$t = 2.5$