5x/2y2 x 8x4/5y2 = ? 5x/2y2 / 8x4/5y2 = ?  

To solve the first problem, multiply the fractions straight across. To multiply the same base, use the exponent rule. Simplify. To solve the second problem, multiply the first fraction by the reciprocal of the second fraction. Apply the exponent rule and simplify.

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(1) `(5x)/(2y^2)*(8x^4)/(5y^2)`

To simplify this expression, we have to multiply the fractions straight across.

=`(5x*8x^4)/(2y^2*5y^2)`

`=(40x*x^4)/(10*y^2*y^2)`

To multiply same base, apply the exponent rule `a^m*a^n=a^(m+n).`

`=(40x^(1+4))/(10y^(2+2))` ` `

`=(40x^5)/(10y^4)`

To simplify this further, cancel the common factor between the numerator and denominator.

`=(4x^5)/y^4`

Therefore, the simplified form of the given...

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(1) `(5x)/(2y^2)*(8x^4)/(5y^2)`

To simplify this expression, we have to multiply the fractions straight across.

=`(5x*8x^4)/(2y^2*5y^2)`

`=(40x*x^4)/(10*y^2*y^2)`

To multiply same base, apply the exponent rule `a^m*a^n=a^(m+n).`

`=(40x^(1+4))/(10y^(2+2))` ` `

`=(40x^5)/(10y^4)`

To simplify this further, cancel the common factor between the numerator and denominator.

`=(4x^5)/y^4`

Therefore, the simplified form of the given expression is `(5x)/(2y^2)*(8x^4)/(5y^2) = (4x^4)/y^4.`

(2) `((5x)/(2y^2))/((8x^4)/(5y^2))`

To divide two fractions, take the reciprocal of the second fraction. Change the operation from division to multiplication.

`= (5x)/(2y^2)*(5y^2)/(8x^4)`

Multiply the fractions straight across.

`=(25xy^2)/(16x^4y^2)`

To divide same base, apply the exponent rule `a^m/a^n=a^(m-n).`

`=25/16 x^(1-4)y^(2-2)`

`=25/16x^(-3)y^0`

Take note that when the exponent is zero, it simplifies to 1.

`=25/16x^(-3)*1`

`=25/16x^(-3)`

To express the x with a positive exponent, apply the rule `a^(-m)=1/a^m.`

`=25/16*1/x^3`

`=25/(16x^3)`

Therefore, the simplified form of the given expression is `((5x)/(2y^2))/((8x^4)/(5y^2))=25/(16x^3).`

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