Question

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which expression is equivalent to $5\sqrt{9t^{7}}$? assume $t$ is greater than or equal to zero. $3t^{3}\cdot\sqrt{t}$, $3t\cdot\sqrt{t}$, $15t^{3}\cdot\sqrt{t}$, $15t\cdot\sqrt{t^{6}}$

Explanation:

Step1: Simplify the square root term

First, simplify $\sqrt{9t^7}$. We can split the radical into $\sqrt{9} \cdot \sqrt{t^7}$. Since $\sqrt{9}=3$, and $\sqrt{t^7}=t^3\sqrt{t}$ (because $t^7 = t^{6+1}=t^6 \cdot t$, and $\sqrt{t^6}=t^3$ for $t\geq0$). So:
$\sqrt{9t^7}=3t^3\sqrt{t}$

Step2: Multiply by the coefficient 5

Multiply the simplified radical by the outer coefficient 5:
$5 \times 3t^3\sqrt{t}=15t^3\sqrt{t}$

Answer:

$15t^3 \cdot \sqrt{t}$