Question

which equation is equivalent to $2^{4x}=8^{x - 3}$?
$2^{4x}=2^{3x - 9}$
$2^{4x}=2^{2x - 6}$
$2^{4x}=2^{2x - 3}$
$2^{4x}=2^{3x - 3}$

Explanation:

Step1: Rewrite 8 as a power of 2

Since \( 8 = 2^3 \), we can rewrite the right - hand side of the equation \( 2^{4x}=8^{x - 3} \) using the property of exponents \((a^m)^n=a^{mn}\). So \( 8^{x - 3}=(2^3)^{x - 3} \).

Step2: Simplify \((2^3)^{x - 3}\)

Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we have \((2^3)^{x - 3}=2^{3\times(x - 3)}\).

Step3: Expand \(3\times(x - 3)\)

Using the distributive property \(a(b - c)=ab - ac\), where \(a = 3\), \(b=x\) and \(c = 3\), we get \(3\times(x - 3)=3x-9\). So \(8^{x - 3}=2^{3x - 9}\).

Step4: Equivalent equation

The original equation \(2^{4x}=8^{x - 3}\) is equivalent to \(2^{4x}=2^{3x - 9}\) after substituting \(8^{x - 3}\) with \(2^{3x - 9}\).

Answer:

\(2^{4x}=2^{3x - 9}\) (the first option among the given options)