Question

which expression is equivalent to \\( \sqrt{49b^4c^7} \\)? assume \\( b \\) and \\( c \\) are greater than or equal to zero.\\( 7b^2c^3 \cdot \sqrt{c} \\) \\( b^2c^3 \cdot \sqrt{7c} \\) \\( b^2 \cdot \sqrt{c} \\) \\( 7b^2c^3 \\)

Explanation:

Step1: Simplify the square root of 49

We know that $\sqrt{49} = 7$ since $7\times7 = 49$.

Step2: Simplify the square root of $b^{4}$

Using the property of exponents $\sqrt{x^{n}}=x^{\frac{n}{2}}$ (for $x\geq0$), for $b^{4}$, we have $\sqrt{b^{4}}=b^{\frac{4}{2}} = b^{2}$.

Step3: Simplify the square root of $c^{7}$

We can rewrite $c^{7}$ as $c^{6}\times c$. Then $\sqrt{c^{7}}=\sqrt{c^{6}\times c}=\sqrt{c^{6}}\times\sqrt{c}$. Since $\sqrt{c^{6}} = c^{\frac{6}{2}}=c^{3}$, we get $\sqrt{c^{7}}=c^{3}\sqrt{c}$.

Step4: Combine the simplified parts

Putting it all together, $\sqrt{49b^{4}c^{7}}=\sqrt{49}\times\sqrt{b^{4}}\times\sqrt{c^{7}} = 7\times b^{2}\times c^{3}\sqrt{c}=7b^{2}c^{3}\sqrt{c}$.

Answer:

$7b^{2}c^{3}\cdot\sqrt{c}$ (the first option)