Question

find the values for a, b, and c that complete the simplification.
\sqrt{x^{12}y^{9}z^{5}} = \sqrt{x^{12} \cdot y^{8} \cdot y \cdot z^{4} \cdot z} = x^{a}y^{b}z^{c}\sqrt{yz}

a = \square

b = \square

c = \square

Explanation:

Step1: Simplify \( \sqrt{x^{12}} \)

Using the property of square roots \( \sqrt{x^n}=x^{\frac{n}{2}} \) (for even \( n \)), for \( x^{12} \), we have \( \sqrt{x^{12}} = x^{\frac{12}{2}} = x^6 \). So \( a = 6 \).

Step2: Simplify \( \sqrt{y^8} \)

Using the same square root property, for \( y^8 \), \( \sqrt{y^8}=y^{\frac{8}{2}} = y^4 \). So \( b = 4 \).

Step3: Simplify \( \sqrt{z^4} \)

For \( z^4 \), \( \sqrt{z^4}=z^{\frac{4}{2}} = z^2 \). So \( c = 2 \).

Answer:

\( a = 6 \), \( b = 4 \), \( c = 2 \)