Question

True or False If \(4^{3}=2^{a},\) then \(a=6 .\) Justify your answer.

Short answer

The statement is true, \(a = 6\).

Step-by-step solution

Express in the same base

Since \(4\) can be expressed as \(2^2\), the left-hand side of the equation becomes \((2^2)^3\). By the rule of exponents, where powers are multiplied when raised by another power, the expression simplifies to \(2^6\). This results in the equation \(2^6 = 2^a\).

Compare the exponents

Since the bases on both sides of the equation are the same \((2)\), the exponents can be equated to each other. This gives us the equation \(6 = a\).

Verify the hypothesis

Looking at the original question, it was hypothesized that \(a = 6\). After calculation, it was found that \(a\) does indeed equal \(6\), therefore the original statement is true.