Question

simplifying radical expressions: multiply and write the product in simplest form.
\sqrt{8x} \cdot 3\sqrt{72x^3}
squares and their roots
11² = 121 21² = 441
12² = 144 22² = 484
13² = 169 23² = 529
14² = 196 24² = 576
15² = 225 25² = 625
16² = 256 26² = 676
17² = 289 27² = 729
18² = 324 28² = 784
19² = 361 29² = 841
20² = 400 30² = 900

Explanation:

Step1: Use radical multiplication rule

The rule for multiplying radicals is \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\) (for \(a\geq0,b\geq0\)). Also, we can multiply the coefficients separately. So, \(\sqrt{8x}\cdot3\sqrt{72x^{3}} = 3\cdot\sqrt{8x\cdot72x^{3}}\)

Step2: Simplify the radicand

First, multiply the coefficients inside the radical: \(8\times72 = 576\). Then, multiply the variables: \(x\cdot x^{3}=x^{4}\). So the radicand becomes \(576x^{4}\), and we have \(3\cdot\sqrt{576x^{4}}\)

Step3: Simplify the square root

We know that \(\sqrt{576}=24\) and \(\sqrt{x^{4}} = x^{2}\) (since \((x^{2})^{2}=x^{4}\)). So \(\sqrt{576x^{4}}=24x^{2}\)

Step4: Multiply the coefficient

Now, multiply the 3 (from the coefficient) with \(24x^{2}\): \(3\times24x^{2}=72x^{2}\)

Answer:

\(72x^{2}\)