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Chapter 4: Problem 79

Perform the operations. $$5 x^{5} y^{10}-\left(2 x y^{2}\right)^{5}+(3 x)^{5} y^{10}$$

Short Answer

Expert verified
The simplified form of given expression \(5 x^{5} y^{10}-(2 x y^{2})^{5}+(3 x)^{5} y^{10}\) is \((5 - 32 + 243) x^{5} y^{10}\), which is \(216 x^{5} y^{10}\).

Step by step solution

01

Distribute exponents in parenthesis

Firstly, distribute the exponent inside the parenthesis to every term inside: \(-(2 x y^{2})^{5} = -(2^{5}x^{5}y^{10}) \) and \((3x)^{5} y^{10} = 3^{5}x^{5} y^{10}\). It is important to remember that \((a b)^{n}=a^{n} b^{n}\) when dealing with a positive integer exponent.
02

Simplify the exponent calculations

In this step, perform the calculation of the constants with exponents. This results in \(2^{5} =32\) and \(3^{5} = 243\), and the equation becomes \(5 x^{5} y^{10} - (32 x^{5} y^{10}) + 243 x^{5} y^{10}\).
03

Simplify equation by combining like terms

The terms \(5 x^{5} y^{10}\), \(- 32 x^{5} y^{10}\) and \(243 x^{5} y^{10}\) are all like terms as they have the same variable to the same exponential power. To simplify, combine the coefficients of these like terms resulting in \((5 -32 +243)x^{5} y^{10}\).

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