Question

multiply the radicals below. ensure your final answer is as simplified as possible:
\sqrt{28} \cdot -2\sqrt{42} = \boxed{}

Explanation:

Step1: Recall radical multiplication rule

The rule for multiplying radicals is $\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}$ (for $a\geq0, b\geq0$), and also we can handle the coefficients separately. So first, multiply the coefficients and the radicals separately. The expression is $\sqrt{28}\cdot(-2\sqrt{42})$. The coefficient part is $1\times(-2)= -2$. The radical part is $\sqrt{28}\cdot\sqrt{42}$.

Step2: Multiply the radicals

Using the radical multiplication rule, $\sqrt{28}\cdot\sqrt{42}=\sqrt{28\times42}$. Now factor 28 and 42: $28 = 4\times7$ and $42 = 6\times7=2\times3\times7$. So $28\times42=(4\times7)\times(2\times3\times7)=4\times2\times3\times7\times7 = 8\times3\times49=24\times49$. So $\sqrt{28\times42}=\sqrt{24\times49}$.

Step3: Simplify the radical

We know that $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ (for $a\geq0, b\geq0$), so $\sqrt{24\times49}=\sqrt{24}\cdot\sqrt{49}$. $\sqrt{49} = 7$, and $\sqrt{24}=\sqrt{4\times6}=\sqrt{4}\cdot\sqrt{6}=2\sqrt{6}$. So $\sqrt{24}\cdot\sqrt{49}=2\sqrt{6}\times7 = 14\sqrt{6}$.

Step4: Multiply with the coefficient

Now we had the coefficient -2 from Step1, so multiply -2 with the result from Step3: $-2\times14\sqrt{6}=-28\sqrt{6}$? Wait, no, wait. Wait in Step2, let's re - calculate $28\times42$: $28\times42=(30 - 2)\times(40+2)=30\times40+30\times2-2\times40 - 2\times2=1200 + 60-80 - 4=1176$. Alternatively, prime factorization: 28 = $2^2\times7$, 42 = $2\times3\times7$. So $28\times42=2^2\times7\times2\times3\times7=2^{3}\times3\times7^{2}$. Then $\sqrt{28\times42}=\sqrt{2^{3}\times3\times7^{2}}=\sqrt{2^{2}\times2\times3\times7^{2}}=\sqrt{2^{2}}\times\sqrt{7^{2}}\times\sqrt{2\times3}=2\times7\times\sqrt{6}=14\sqrt{6}$. Then the coefficient part is -2, so total is $-2\times14\sqrt{6}=-28\sqrt{6}$? Wait, no, wait the original expression is $\sqrt{28}\times(-2\sqrt{42})=(-2)\times\sqrt{28}\times\sqrt{42}=(-2)\times\sqrt{28\times42}=(-2)\times\sqrt{1176}$. Now $\sqrt{1176}=\sqrt{49\times24}=\sqrt{49}\times\sqrt{24}=7\times\sqrt{4\times6}=7\times2\sqrt{6}=14\sqrt{6}$. So $(-2)\times14\sqrt{6}=-28\sqrt{6}$? Wait, no, wait 28×42: 28×40 = 1120, 28×2 = 56, so 1120+56 = 1176. Then 1176 divided by 49 is 24, so $\sqrt{1176}=\sqrt{49\times24}=7\sqrt{24}=7\times2\sqrt{6}=14\sqrt{6}$. Then multiply by -2: -2×14√6 = -28√6? Wait, but let's check again. Wait, $\sqrt{28}=2\sqrt{7}$, $\sqrt{42}=\sqrt{6\times7}=\sqrt{6}\times\sqrt{7}$. So $\sqrt{28}\times\sqrt{42}=2\sqrt{7}\times\sqrt{6}\times\sqrt{7}=2\times7\times\sqrt{6}=14\sqrt{6}$. Then multiply by -2: -2×14√6 = -28√6. Yes, that's correct.

Answer:

$-28\sqrt{6}$