How do you simplify #(2p^3q^2)/(8p^4q) div (4pq^2)/(16p^4)#?

Question

Abigail Allen · Accepted Answer

#((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))=p^2/q#

We start off with what's given:

#((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))#

Recall: #(a/b)/(c/d)=(a/b)/(c/d)*(d/c)/(d/c)=a/b*d/c#

So,

#((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))=((2p^3q^2)/(8p^4q))*((16p^4)/(4pq^2))#

Recall: #a^5/a^3=a^5*a^-3=a^(5-3)=a^2#

So,

#((2p^3q^2)/(8p^4q))((16p^4)/(4pq^2))##=##((p^3q^2)/(4p^4q))((4p^4)/(pq^2))#

The #((p^3q^2)/(4p^4q))# part becomes:

#((p^3q^2)/(4p^4q))=(p^3q^2)(4^(-1)p^(-4)q^(-1))=4^(-1)p^(3-4)q^(2-1)=4^(-1)p^(-1)q^1#

#=q/(4p)#

The #((4p^4)/(pq^2))# part becomes:

#((4p^4)/(pq^2))=(4p^4)(p^(-1)q^(-2))=4p^(4-1)q^(-2)=4p^3q^(-2)#

#=(4p^3)/q^2#

So, our original setup was:

#((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))=((2p^3q^2)/(8p^4q))*((16p^4)/(4pq^2))=((p^3q^2)/(4p^4q))((4p^4)/(pq^2))#

#=(q/(4p))((4p^3)/q^2)=(4p^3q)/(4pq^2)=p^2/q#

Julia Adams · Answer

The answer is #p^2/q#.

#(2p^3q^2)/(8p^4q)##-:##(4pq^2)/(16p^4)# Reduce the numerical part of each fraction. #(cancel2^1p^3q^2)/(cancel8^4p^4q)##-:##(cancel4^1pq^2)/(cancel16^4p^4)# = #(p^3q^2)/(4p^4q)##-:##(pq^2)/(4p^4)# = Simplify each fraction. Use the exponent rule #x^m/x^n=x^(m-n)# for each fraction. #(p^3q^2)/(4p^4q)=(p^(3-4)q^(2-1))/(4)=(p^(-1)q)/4# #(pq^2)/(4p^4)=(p^(1-4)q^2)/(4)=(p^-3q^2)/4# Use exponent rule #x^(-m)=1/(x^m)# for each fraction. #(q)/(4p)-:(q^2)/(4p^3)# In order to divide fractions, invert the second fraction to get its reciprocal, and then multiply. #(q)/(4p)xx(4p^3)/(q^2)# = #(4p^3q)/(4pq^2)# = Cancel the #4#. #(cancel4^1p^3q)/(cancel4^1pq^2)# Apply the exponent rules #x^m/x^n=x^(m-n)# and #x^(-m)=1/(x^m)#. #(p^(3-1)q^(1-2))=p^2q^(-1)=(p^2)/q#

Eli Assouline · Answer

undefined To simplify the expression (2p^3q^2)/(8p^4q) ÷ (4pq^2)/(16p^4), we can follow these steps:

1. Rewrite the division as multiplication by taking the reciprocal of the second fraction: (2p^3q^2)/(8p^4q) × (16p^4)/(4pq^2).
2. Simplify the numerator by multiplying the coefficients and combining the variables with the same base: (2 × 16) × (p^3 × p^4) × (q^2)/(8 × 4 × p^4 × q × q^2).
3. Simplify the denominator by multiplying the coefficients and combining the variables with the same base: (2 × 16) × (p^3 × p^4) × (q^2)/(8 × 4 × p^4 × q × q^2).
4. Cancel out common factors between the numerator and denominator: (2 × 16)/(8 × 4) × (p^3 × p^4)/(p^4) × (q^2)/(q × q^2).
5. Simplify the remaining expression: 32/32 × p^(3+4)/p^4 × q^(2-1)/q^2.
6. Further simplify the expression: 1 × p^7/p^4 × q/q^2.
7. Combine the variables with the same base by subtracting the exponents: p^(7-4) × q^(1-2).
8. Simplify the exponents: p^3/q.
9. The simplified expression is (p^3)/q.