Question

1. which expression is equivalent to $7\sqrt{45}$?

a. $12\sqrt{3}$
b. $35\sqrt{3}$
c. $10\sqrt{5}$
d. $21\sqrt{5}$

Explanation:

Step1: Simplify the square root

First, factor 45 into its prime factors. We know that \(45 = 9\times5\), and \(9\) is a perfect square (\(9 = 3^2\)). So, we can rewrite \(\sqrt{45}\) as \(\sqrt{9\times5}\).
Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)), we get \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}\).
Since \(\sqrt{9} = 3\), this simplifies to \(3\sqrt{5}\).

Step2: Multiply by the coefficient

Now, we have the original expression \(7\sqrt{45}\). Substituting the simplified square root from Step 1, we get \(7\times3\sqrt{5}\).
Multiplying the coefficients \(7\) and \(3\), we have \(7\times3 = 21\). So, the expression simplifies to \(21\sqrt{5}\).

Answer:

D. \(21\sqrt{5}\)