Question

solve: $x^4 = 16$

Explanation:

Step1: Rewrite the equation

Rewrite \(16\) as \(2^4\) and also consider the negative root since even - powered equations can have negative solutions. So the equation \(x^{4}=16\) can be written as \(x^{4}-16 = 0\). Using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), we first let \(a=x^{2}\) and \(b = 4\), then \(x^{4}-16=(x^{2}+4)(x^{2}-4)=0\). And then we can factor \(x^{2}-4\) further as \((x + 2)(x - 2)\) since \(x^{2}-4\) is also a difference of squares. So the fully factored form is \((x^{2}+4)(x + 2)(x - 2)=0\).

Step2: Solve for x

Set each factor equal to zero:

• For \(x - 2=0\), we get \(x = 2\).
• For \(x + 2=0\), we get \(x=-2\).
• For \(x^{2}+4=0\), we have \(x^{2}=-4\), and the solutions for this are \(x = 2i\) and \(x=-2i\) (where \(i\) is the imaginary unit with \(i^{2}=-1\)). But if we are only considering real - valued solutions, we focus on the factors \((x + 2)\) and \((x - 2)\).

Answer:

The real solutions of the equation \(x^{4}=16\) are \(x = 2\) and \(x=-2\). If we consider complex solutions, the solutions are \(x = 2\), \(x=-2\), \(x = 2i\), and \(x=-2i\).