To factor the expression [tex]\( 4m^6 - 16p^{10} \)[/tex], we'll follow a step-by-step approach:
1. Identify Common Factors:
Notice that both terms in the expression [tex]\( 4m^6 - 16p^{10} \)[/tex] have a common factor of 4. Thus, we can factor out this common factor first:
[tex]\[
4m^6 - 16p^{10} = 4(m^6 - 4p^{10})
\][/tex]
2. Recognize the Difference of Squares:
Next, we look inside the parentheses. Notice that [tex]\( m^6 - 4p^{10} \)[/tex] is a difference of squares. Let's rewrite [tex]\( m^6 \)[/tex] as [tex]\( (m^3)^2 \)[/tex] and [tex]\( 4p^{10} \)[/tex] as [tex]\( (2p^5)^2 \)[/tex]:
[tex]\[
m^6 - 4p^{10} = (m^3)^2 - (2p^5)^2
\][/tex]
3. Apply the Difference of Squares Formula:
The difference of squares formula states that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
Applying this to our expression:
[tex]\[
(m^3)^2 - (2p^5)^2 = (m^3 - 2p^5)(m^3 + 2p^5)
\][/tex]
4. Combine Factors:
Substituting this result back into the expression we factored out in step 1:
[tex]\[
4(m^6 - 4p^{10}) = 4((m^3)^2 - (2p^5)^2) = 4(m^3 - 2p^5)(m^3 + 2p^5)
\][/tex]
Thus, the fully factored form of the original expression [tex]\( 4m^6 - 16p^{10} \)[/tex] is:
[tex]\[
4(m^3 - 2p^5)(m^3 + 2p^5)
\][/tex]
