Question

follow the instructions below.
write ((2x)^4) without exponents.
((2x)^4 = square)
fill in the blanks.
((2x)^4 = square x^{square})

Explanation:

First Part: Write \((2x)^4\) without exponents

Step1: Recall the power of a product rule

The power of a product rule states that \((ab)^n = a^n\times b^n\). For \((2x)^4\), we can apply this rule where \(a = 2\), \(b=x\) and \(n = 4\). So \((2x)^4=2^4\times x^4\).

Step2: Calculate \(2^4\)

We know that \(2^4=2\times2\times2\times2 = 16\) and \(x^4=x\times x\times x\times x\). But since we need to write without exponents for the first blank (expanding the product), we also can expand \((2x)^4\) as \((2x)\times(2x)\times(2x)\times(2x)\). Let's calculate that:
First, multiply the coefficients and the variables step by step. \((2x)\times(2x)=4x^2\), then \((4x^2)\times(2x) = 8x^3\), then \((8x^3)\times(2x)=16x^4\). But if we expand \((2x)^4\) as a product of four \((2x)\) terms: \((2x)\times(2x)\times(2x)\times(2x)=2\times x\times2\times x\times2\times x\times2\times x\). Then we can multiply the coefficients: \(2\times2\times2\times2 = 16\) and the variables: \(x\times x\times x\times x=x^4\). But if we want to write without exponents (expanding the \(x\) terms too), we have \(2\times2\times2\times2\times x\times x\times x\times x=16\times x\times x\times x\times x\). But maybe the first blank is expecting the expanded form as a product of the factors. Wait, actually, the first part says "write without exponents", so we can expand \((2x)^4\) as \((2x)\times(2x)\times(2x)\times(2x)\), but let's check the second part too. Wait, maybe the first blank (the first equation) is to expand \((2x)^4\) as a product. Let's do that:

\((2x)^4=(2x)\times(2x)\times(2x)\times(2x)\). But also, using the power of a product, we can compute the numerical coefficient and the variable part. Wait, maybe the first blank is \(16x^4\) but no, wait the problem says "without exponents". Wait, maybe I misread. Wait, the first task is "Write \((2x)^4\) without exponents". So we need to expand it completely, so \((2x)^4 = 2\times2\times2\times2\times x\times x\times x\times x\), but that's a bit long. Alternatively, using the power of a product, \(2^4 = 16\) and \(x^4=x\times x\times x\times x\), so \((2x)^4=16\times x\times x\times x\times x\). But maybe the first blank is \(16x^4\) but no, the problem says "without exponents". Wait, maybe the first part is to expand \((2x)^4\) as a product of four \((2x)\) terms: \((2x)\times(2x)\times(2x)\times(2x)\). Let's confirm:

\((2x)^4\) means multiplying \(2x\) by itself 4 times. So \((2x)\times(2x)\times(2x)\times(2x)\).

Second Part: Fill in the blanks \((2x)^4=\square x^{\square}\)

Step1: Apply the power of a product rule

Using the power of a product rule \((ab)^n=a^n b^n\), for \((2x)^4\), we have \(a = 2\), \(b = x\), \(n=4\). So \((2x)^4=2^4\times x^4\).

Step2: Calculate \(2^4\)

We know that \(2^4=2\times2\times2\times2 = 16\). So \((2x)^4 = 16x^4\).

Answer:

(First Part - Write without exponents):
\((2x)^4=\boldsymbol{(2x)\times(2x)\times(2x)\times(2x)}\) (or \(2\times2\times2\times2\times x\times x\times x\times x\))