Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Mar 08, 2024

Solve the equation $\frac{16^{1/3}}{4^{x/3}}=1$ to find the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $\frac{16^{\frac{1}{3}}}{4^{\frac{x}{3}}}=1$ .
2. We need to solve for the variable $x$ .

STEP 2

Recognize that $16$ and $4$ are both powers of $2$ .
$16 = 2^4$ $4 = 2^2$

STEP 3

Rewrite the given equation using the base $2$ .
$\frac{(2^4)^{\frac{1}{3}}}{(2^2)^{\frac{x}{3}}}=1$

STEP 4

Apply the power of a power rule, which states that $(a^m)^n = a^{mn}$ .
$\frac{2^{4 \cdot \frac{1}{3}}}{2^{2 \cdot \frac{x}{3}}}=1$

STEP 5

Simplify the exponents.
$\frac{2^{\frac{4}{3}}}{2^{\frac{2x}{3}}}=1$

STEP 6

Since the bases are the same, we can equate the exponents.
$\frac{4}{3} - \frac{2x}{3} = 0$

STEP 7

Multiply through by 3 to clear the denominators.
$4 - 2x = 0$

STEP 8

Isolate the variable $x$ .
$2x = 4$

STEP 9

Divide both sides by 2 to solve for $x$ .
$x = \frac{4}{2}$

STEP 10

Simplify the fraction.
$x = 2$
The solution to the equation is $x=2$ .

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