Question

question
use multiplication to fully expand the expression below.
$(xy)^2$
answer attempt 1 out of 2
press the · button or type the * symbol on your keyboard to represent multiplication. using × for multiplication is inappropriate when x may be used as a variable.
expanded form:

Explanation:

Step1: Recall the exponent rule

The exponent rule for a power of a product states that \((ab)^n = a^n \cdot b^n\). For \((xy)^2\), we can apply this rule. Also, \(a^2=a\times a\) by the definition of exponent (where the exponent indicates the number of times the base is multiplied by itself).

Step2: Expand the expression

First, using the power of a product rule, \((xy)^2 = x^2 \cdot y^2\). Then, expanding \(x^2\) and \(y^2\) using the definition of exponent, we get \(x^2 = x\times x\) and \(y^2 = y\times y\). So, substituting back, \((xy)^2=(x\times x)\times(y\times y)\) or more simply, since multiplication is associative and commutative, \((xy)^2 = x\times y\times x\times y\) (we can also write it as \(x\times x\times y\times y\) but the order of multiplication doesn't matter here). But following the exponent rule step - by - step, we can also think of \((xy)^2=(xy)\times(xy)\) (because an exponent of 2 means the base is multiplied by itself 2 times), and then using the commutative property of multiplication ( \(a\times b = b\times a\)) and associative property ( \((a\times b)\times c=a\times(b\times c)\)), we can rewrite \((xy)\times(xy)\) as \(x\times y\times x\times y\) or \(x\times x\times y\times y\). However, a more straightforward expansion using the power of a product and then the definition of exponent is:
\((xy)^2=(x\times y)\times(x\times y)\) (since \((xy)^2\) means \(xy\) multiplied by \(xy\))
Then, using the associative and commutative properties of multiplication, we can rearrange the factors:
\((x\times y)\times(x\times y)=x\times x\times y\times y\) or \(x\times y\times x\times y\) (both are correct, but we can also write it as \(x\times x\times y\times y\) which is equivalent to \(x^{2}y^{2}\) in expanded form with multiplication signs. But the problem asks for the expanded form using multiplication, so we can write it as \(x\times x\times y\times y\) or \(x\times y\times x\times y\). But the most direct expansion from \((xy)^2=(xy)\times(xy)\) is \(x\times y\times x\times y\) (or \(x\times x\times y\times y\)).

Answer:

\(x\times x\times y\times y\) (or \(x\times y\times x\times y\))