Question

Complete the expression subtracting the radicals
14 sqrt3 - 4 sqrt⬜ = 2 sqrt⬜

Answer 1

What is the product of the following in simplest radical form?

√85× 3√32

Answer 2

To complete the expression \( 14\sqrt{3} - 4\sqrt{\square} = 2\sqrt{\square} \), we first want to isolate the radical term.

Starting from the equation:

\[
14\sqrt{3} - 4\sqrt{\square} = 2\sqrt{\square}
\]

we can add \( 4\sqrt{\square} \) to both sides:

\[
14\sqrt{3} = 2\sqrt{\square} + 4\sqrt{\square}
\]

This simplifies to:

\[
14\sqrt{3} = 6\sqrt{\square}
\]

Next, we isolate \( \sqrt{\square} \) by dividing both sides by \( 6 \):

\[
\sqrt{\square} = \frac{14\sqrt{3}}{6}
\]

Simplifying the right side:

\[
\sqrt{\square} = \frac{7\sqrt{3}}{3}
\]

Now, to get \( \square \), we square both sides:

\[
\square = \left(\frac{7\sqrt{3}}{3}\right)^2
\]

Calculating the square:

\[
\square = \frac{49 \cdot 3}{9} = \frac{147}{9} = \frac{49}{3}
\]

Thus, completing the expression yields:

\[
\square = \frac{49}{3}
\]

So the full expression is:

\[
14\sqrt{3} - 4\sqrt{\frac{49}{3}} = 2\sqrt{\frac{49}{3}}
\]

Answer 3

To find the product of \( \sqrt{85} \times 3\sqrt{32} \), we can first separate the constants from the radicals:

\[
\sqrt{85} \times 3\sqrt{32} = 3 \times \sqrt{85} \times \sqrt{32}
\]

Next, we can use the property of radicals that states \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \):

\[
3 \times \sqrt{85 \times 32}
\]

Now we compute \( 85 \times 32 \):

\[
85 \times 32 = 2720
\]

Now, we have:

\[
3 \sqrt{2720}
\]

Next, we simplify \( \sqrt{2720} \). First, we factor \( 2720 \).

Finding the prime factorization of \( 2720 \):
- \( 2720 \div 10 = 272 \)
- \( 272 = 2^4 \times 17 \) (since \( 272 = 16 \times 17 \))

So the prime factorization gives:

\[
2720 = 10 \times 272 = 10 \times 16 \times 17 = 2^4 \times 5 \times 17
\]

Now, we can rewrite \( \sqrt{2720} \):

\[
\sqrt{2720} = \sqrt{2^4 \times 5 \times 17} = \sqrt{(2^2)^2 \times 5 \times 17} = 2^2 \sqrt{5 \times 17} = 4\sqrt{85}
\]

Now substituting back, we have:

\[
3 \sqrt{2720} = 3 \times 4 \sqrt{85} = 12\sqrt{85}
\]

Thus, the product \( \sqrt{85} \times 3\sqrt{32} \) in simplest radical form is:

\[
\boxed{12\sqrt{85}}
\]