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Multiply. State the product in simplest form.

[tex]\[ \frac{45a^3}{a^2-4} \cdot \frac{a^2+2a}{30a^2} \][/tex]

A. [tex]\(\frac{3a}{a-2}\)[/tex]

B. [tex]\(\frac{9a^2}{6a-12}\)[/tex]

C. [tex]\(\frac{3a^2}{-4}\)[/tex]

D. [tex]\(\frac{3a^2}{2a-4}\)[/tex]


Sagot :

Certainly! Let's multiply the given expressions step-by-step and simplify the result.

We start with the expression:
[tex]\[ \frac{45 a^3}{a^2 - 4} \cdot \frac{a^2 + 2a}{30 a^2} \][/tex]

Step 1: Multiply the Numerators

The numerators of the two fractions are:
[tex]\[ 45 a^3 \quad \text{and} \quad a^2 + 2a \][/tex]

Multiplying these together:
[tex]\[ 45 a^3 \cdot (a^2 + 2a) = 45 a^3(a^2 + 2a) = 45 a^5 + 90 a^4 \][/tex]

Step 2: Multiply the Denominators

The denominators of the fractions are:
[tex]\[ a^2 - 4 \quad \text{and} \quad 30 a^2 \][/tex]

Multiplying these together:
[tex]\[ (a^2 - 4) \cdot 30 a^2 = 30 a^2 (a^2 - 4) = 30 a^4 - 120 a^2 \][/tex]

Step 3: Combine the Results

Combining the multiplied numerators and denominators, we get:
[tex]\[ \frac{45 a^5 + 90 a^4}{30 a^4 - 120 a^2} \][/tex]

Step 4: Simplify the Fraction

To simplify, we should look for common factors in the numerator and the denominator. Let's rewrite and factor both numerator and denominator:

Numerator: [tex]\(45 a^5 + 90 a^4 = 45 a^4 (a + 2)\)[/tex]

Denominator: [tex]\(30 a^4 - 120 a^2 = 30 a^2 (a^2 - 4) = 30 a^2 (a + 2)(a - 2)\)[/tex]

So the expression becomes:
[tex]\[ \frac{45 a^4 (a + 2)}{30 a^2 (a + 2)(a - 2)} \][/tex]

We can cancel out common factors from the numerator and the denominator. Specifically, [tex]\(45 a^4\)[/tex] and [tex]\(30 a^2\)[/tex] share a common factor:

[tex]\[ \frac{3 \cdot 15 a^4 (a + 2)}{2 \cdot 15 a^2 (a + 2)(a - 2)} = \frac{3 a^2}{2 (a - 2)} \][/tex]

Thus, the product in simplest form is:
[tex]\[ \frac{3 a^2}{2 (a - 2)} \][/tex]

From the options given, this matches with:
[tex]\[ \frac{3 a^2}{2 a - 4} \][/tex]

So, the simplest form of the product is:
[tex]\[ \boxed{\frac{3 a^2}{2 (a-2)}} \][/tex]