Question

which expression is equivalent to (sqrt6{32x^5 y^{10} z^{16}})?
(2y^5 z^{10})
(2xy^2 z^3)
(6y^5 z^{10})
(6xy^2 z^3)

Explanation:

Step1: Factor the radicand

First, factor each part of the radicand \(32x^5y^{10}z^{15}\) into perfect powers of the index (which is 5 here).

• For the coefficient: \(32 = 2^5\)
• For \(x\): \(x^5\) is already a perfect 5th power.
• For \(y\): \(y^{10}=(y^2)^5\) (since \(2\times5 = 10\))
• For \(z\): \(z^{15}=(z^3)^5\) (since \(3\times5 = 15\))

So, we can rewrite the radicand as \(2^5\times x^5\times(y^2)^5\times(z^3)^5\).

Step2: Apply the nth root property

The nth root of a product is the product of the nth roots, and \(\sqrt[n]{a^n}=a\) (for real numbers \(a\) when \(n\) is odd, which it is here, \(n = 5\)). So,

$$ \sqrt[5]{2^5\times x^5\times(y^2)^5\times(z^3)^5}=\sqrt[5]{2^5}\times\sqrt[5]{x^5}\times\sqrt[5]{(y^2)^5}\times\sqrt[5]{(z^3)^5} $$

Using the property \(\sqrt[n]{a^n}=a\), this simplifies to:

$$ 2\times x\times y^2\times z^3 $$

Which is \(2xy^2z^3\).

Answer:

\(2xy^2z^3\) (matching the second option, e.g., if the second option is labeled as, say, B. \(2xy^2z^3\), then the answer would be B. \(2xy^2z^3\))