How do you simplify #(\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }#?

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Answer 1 · 2018-05-03T21:55:54

Here's how it's done

Cancel common factor #x#
#=x^6y^3#
#\=(x^6y^3\)^{\frac{1}{3}}#

Apply the exponent rule
#=\(x^6\)^{\frac{1}{3}}\(y^3\)^{\frac{1}{3}}#

Final solution
#=x^2y#

Answer 2 · 2018-05-04T00:42:05

#" "#
#color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y#

#" "#
Given the expression: #color(red)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }#

#"Exponents formula required: "#

#color(blue)( a^m/a^n = a^(m-n)#

#color(blue)( a^m*a^n = a^(m+n)#

#color(blue)(( a^m)^(1/n) = a^(m/n)#

#color(green)("Step 1:"#

Rewrite the given expression: #color(brown)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }# as

#[[(x^7*y^3)^(1/3)]]/[[x^(1/3)]#

Rewrite as:

#[(x^7)^(1/3)*(y^3)^(1/3)]/x^(1/3)#

#color(green)("Step 2:"#

Use: #color(blue)(( a^m)^(1/n) = a^(m/n)#

Simplify:

#(x^(7/3)*y^(3/3))/x^(1/3)#

#color(green)("Step 3:"#

#(x^(7/3)*y^1)/x^(1/3)#

#color(green)("Step 4:"#

Use: #color(blue)( a^m/a^n = a^(m-n)#

#[x^(7/3)-x^(1/3)]*y#

#x^(6/3)*y#

#rArr x^2*y#

Hence,

#color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y#

Hope it helps.