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Which expression is equivalent to the given expression?
[tex]\[ \left(-3x^3 y^2\right)\left(5x^5 y^4\right)^2 \][/tex]

A. [tex]\(-30 x^{28} y^{18}\)[/tex]

B. [tex]\(-75 x^{50} y^{16}\)[/tex]

C. [tex]\(-75 x^{13} y^{10}\)[/tex]

D. [tex]\(-30 x^{10} y^8\)[/tex]

Sagot :

GinnyAnsweridn
To determine which expression is equivalent to the given expression, [tex]\( \left(-3 x^3 y^2\right)\left(5 x^5 y^4\right)^2 \)[/tex], we will simplify the expression step-by-step.

1. Simplify the inner expression:
[tex]\[ \left(5 x^5 y^4\right)^2 \][/tex]
When we raise a product to a power, we raise each factor to that power:
[tex]\[ (5 x^5 y^4)^2 = 5^2 \cdot (x^5)^2 \cdot (y^4)^2 \][/tex]
Which simplifies to:
[tex]\[ = 25 \cdot x^{10} \cdot y^8 \][/tex]

2. Multiply by the outer expression:
[tex]\[ \left(-3 x^3 y^2\right) \cdot (25 x^{10} y^8) \][/tex]
Apply the distributive property:
[tex]\[ -3 \cdot 25 \cdot x^3 \cdot x^{10} \cdot y^2 \cdot y^8 \][/tex]
Simplify the coefficients:
[tex]\[ -3 \times 25 = -75 \][/tex]
Apply the properties of exponents:
[tex]\[ x^3 \cdot x^{10} = x^{3+10} = x^{13} \][/tex]
[tex]\[ y^2 \cdot y^8 = y^{2+8} = y^{10} \][/tex]

Putting it all together, we obtain:
[tex]\[ -75 x^{13} y^{10} \][/tex]

Thus, the expression that is equivalent to [tex]\( \left(-3 x^3 y^2\right)\left(5 x^5 y^4\right)^2 \)[/tex] is:
[tex]\[ -75 x^{13} y^{10} \][/tex]

So, the correct answer is:
[tex]\[ -75 x^{13} y^{10} \][/tex]
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