Question Solve $2.4^{-3t} \cdot 2.4^{-5} = 2.4^{11}$ for $t$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $\left(2.4^{-3 t}\right)\left(2.4^{-5}\right)=\left(2.4^{11}\right)$ .
2. We need to solve for the variable $t$ .
3. The properties of exponents will be used to solve the equation.

STEP 2

Use the property of exponents that states when you multiply two exponents with the same base, you can add the exponents.
$a^m \cdot a^n = a^{m+n}$

STEP 3

Apply the property of exponents to the left-hand side of the equation.
$\left(2.4^{-3 t}\right)\left(2.4^{-5}\right) = 2.4^{-3t - 5}$

STEP 4

Now, we have the equation in the form $2.4^{-3t - 5} = 2.4^{11}$ .

STEP 5

Since the bases are the same and the equation is an equality, we can set the exponents equal to each other.
$-3t - 5 = 11$

STEP 6

Add 5 to both sides of the equation to isolate the term containing $t$ .
$-3t - 5 + 5 = 11 + 5$

STEP 7

Simplify both sides of the equation.
$-3t = 16$

STEP 8

Divide both sides of the equation by $-3$ to solve for $t$ .
$t = \frac{16}{-3}$

STEP 9

Simplify the fraction to find the value of $t$ .
$t = -\frac{16}{3}$
$t$ is equal to $-\frac{16}{3}$ .

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