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Math  /  Algebra

Question 8. −827x10y5=y17273=93=33−33\begin{array}{l}\text { 8. }-8 \sqrt{27 x^{10} y^{5}=y^{17}} \\ \frac{27}{3}=\frac{9}{3}=\frac{3}{3}-3 \sqrt{3}\end{array}

Studdy Solution

STEP 1

1. The problem involves simplifying the expression −827x10y5-8 \sqrt{27x^{10}y^5}.
2. The expression under the square root can be simplified by factoring out perfect squares.
3. The handwritten calculations suggest breaking down the numbers and variables into simpler components.

STEP 2

1. Simplify the expression under the square root by factoring.
2. Simplify the square root expression.
3. Multiply the simplified square root by −8-8.

STEP 3

Factor the expression under the square root, 27x10y527x^{10}y^5, into perfect squares:
27=33 27 = 3^3 x10=(x5)2 x^{10} = (x^5)^2 y5=y4⋅y=(y2)2⋅y y^5 = y^4 \cdot y = (y^2)^2 \cdot y
Thus, the expression becomes:
27x10y5=(32⋅3)⋅(x5)2⋅(y2)2⋅y 27x^{10}y^5 = (3^2 \cdot 3) \cdot (x^5)^2 \cdot (y^2)^2 \cdot y

STEP 4

Simplify the square root:
27x10y5=(32)⋅3⋅(x5)2⋅(y2)2⋅y \sqrt{27x^{10}y^5} = \sqrt{(3^2) \cdot 3 \cdot (x^5)^2 \cdot (y^2)^2 \cdot y}
=3⋅x5⋅y2⋅3y = 3 \cdot x^5 \cdot y^2 \cdot \sqrt{3y}

STEP 5

Multiply the simplified square root by −8-8:
−8⋅(3⋅x5⋅y2⋅3y) -8 \cdot (3 \cdot x^5 \cdot y^2 \cdot \sqrt{3y})
=−24x5y23y = -24x^5y^2\sqrt{3y}
The simplified expression is:
−24x5y23y \boxed{-24x^5y^2\sqrt{3y}}

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