Simplify the expression start fraction 2 a left parenthesis a squared b right parenthesis cubed over a superscript 4 baseline b superscript 5 baseline end fraction.

Question

(1 point)
Responses

start fraction 2 a cubed over b squared end fraction
Image with alt text: start fraction 2 a cubed over b squared end fraction

start fraction 2 a squared over b squared end fraction
Image with alt text: start fraction 2 a squared over b squared end fraction

start fraction 8 a squared over b end fraction
Image with alt text: start fraction 8 a squared over b end fraction

start fraction 8 a cubed over b end fraction

Answer 1

To simplify the expression \(\frac{2a (a^2b)^3}{a^4b^5}\), we start by simplifying the numerator.

1. Calculate \((a^2b)^3\): \[ (a^2b)^3 = a^{2 \cdot 3} b^{1 \cdot 3} = a^6b^3 \]

2. Multiply by \(2a\): \[ 2a(a^6b^3) = 2a^1 a^6 b^3 = 2a^{1+6}b^3 = 2a^7b^3 \]

Now, we rewrite the expression using this result: \[ \frac{2a^7b^3}{a^4b^5} \]

3. Simplify the fraction:
   • For the \(a\) terms: \[ \frac{a^7}{a^4} = a^{7-4} = a^3 \]
   • For the \(b\) terms: \[ \frac{b^3}{b^5} = b^{3-5} = b^{-2} \]

Putting it all together, we get: \[ \frac{2a^3}{b^2} \]

Thus, the simplified expression is: \[ \frac{2a^3}{b^2} \]

The correct answer is: \[ \frac{2a^3}{b^2} \]