Math

QuestionSimplify the expression: (5)3x7(yz)3(4)2x5y7z2\frac{(-5)^{3} x^{7}(y z)^{3}}{(4)^{2} x^{5} y^{7} z^{2}}

Studdy Solution

STEP 1

Assumptions1. We are given the expression (5)3x7(yz)3(4)x5y7z\frac{(-5)^{3} x^{7}(y z)^{3}}{(4)^{} x^{5} y^{7} z^{}}. . We assume that all variables are real numbers.
3. The rules of exponents apply, where aman=am+na^{m}a^{n} = a^{m+n}, (am)n=amn(a^{m})^{n} = a^{mn}, and aman=amn\frac{a^{m}}{a^{n}} = a^{m-n}.

STEP 2

First, simplify the expression by dealing with the numerical values and the variables separately.(5)x7(yz)(4)2x5y7z2=(5)(4)2x7x5(yz)y7z2\frac{(-5)^{} x^{7}(y z)^{}}{(4)^{2} x^{5} y^{7} z^{2}} = \frac{(-5)^{}}{(4)^{2}} \cdot \frac{x^{7}}{x^{5}} \cdot \frac{(y z)^{}}{y^{7} z^{2}}

STEP 3

implify the numerical part of the expression.
(5)3()2=12516\frac{(-5)^{3}}{()^{2}} = \frac{-125}{16}

STEP 4

implify the part of the expression with the variable xx using the rule aman=amn\frac{a^{m}}{a^{n}} = a^{m-n}.
x7x=x7=x2\frac{x^{7}}{x^{}} = x^{7-} = x^{2}

STEP 5

implify the part of the expression with the variables yy and zz. Here, we first use the rule (am)n=amn(a^{m})^{n} = a^{mn} to expand (yz)3(y z)^{3}, and then use the rule aman=amn\frac{a^{m}}{a^{n}} = a^{m-n} to simplify.
(yz)3y7z2=y3z3y7z2=y37z32=y4z1\frac{(y z)^{3}}{y^{7} z^{2}} = \frac{y^{3} z^{3}}{y^{7} z^{2}} = y^{3-7} z^{3-2} = y^{-4} z^{1}

STEP 6

Combine all the simplified parts of the expression to get the final answer.
12516x2y4z1=12516x2y4z\frac{-125}{16} \cdot x^{2} \cdot y^{-4} z^{1} = -\frac{125}{16} x^{2} y^{-4} zSo, (5)3x(yz)3(4)2x5yz2=12516x2y4z\frac{(-5)^{3} x^{}(y z)^{3}}{(4)^{2} x^{5} y^{} z^{2}} = -\frac{125}{16} x^{2} y^{-4} z.

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