what is the simplified form of $sqrt{10,000x^{64}}$?\n$5000x^{32}$\n$5000x^{8}$\n$100x^{8}$\n$100x^{32}$

Explanation:

Step1: Break down 10000

We know that $10000 = 100\times100=100^{2}$, so $\sqrt{10000x^{64}}=\sqrt{100^{2}x^{64}}$.

Step2: Apply square - root property

According to the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ ($a = 100^{2}$, $b=x^{64}$), we have $\sqrt{100^{2}x^{64}}=\sqrt{100^{2}}\cdot\sqrt{x^{64}}$.

Step3: Calculate square - roots

Since $\sqrt{100^{2}} = 100$ and $\sqrt{x^{64}}=x^{32}$ (because $\sqrt{x^{n}}=x^{\frac{n}{2}}$ and $\frac{64}{2}=32$), then $\sqrt{100^{2}}\cdot\sqrt{x^{64}}=100x^{32}$.

Answer:

$100x^{32}$