Question

Complete the expression adding the radicals.(1 point)

(blank)√5 + 3 √45 = 13√(blank)

Answer 1

Complete the expression subtracting the radicals.
(blank)√(blank) - (blank)√(blank) = (blank)√(blank)

Answer 2

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

8√5–⋅3√32−−


(1 point)

(blank)√(blank)

Answer 3

Maura is creating a quilt using right triangles. A large right triangle with a side length of 40√12−−
centimeters has smaller right triangles with side lengths of 5√3–
centimeters along it. How many smaller lengths are running along the longer length?(1 point)
(blank)

Answer 4

Consider the expression 47/3 + √121−−−
. Will the sum of the two rational numbers produce a rational number? Explain your reasoning in 1–2 sentences.

Answer 5

Prove that the product of the two rational numbers 36−−√
and 5/3
is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1–2 sentences.(2 points)

Answer 6

To simplify the expression \( 3 \sqrt{45} \), we can start by factoring \( \sqrt{45} \):

\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \sqrt{5} = 3\sqrt{5}
\]

Now we substitute this back into the expression:

\[
3 \sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5}
\]

Now we can rewrite the equation as follows:

\[
\text{(blank)} \sqrt{5} + 9 \sqrt{5} = 13 \sqrt{(blank)}
\]

Letting \( x \) represent the blank in the first part:

\[
x \sqrt{5} + 9 \sqrt{5} = (x + 9) \sqrt{5}
\]

Setting it equal to the right-hand side:

\[
(x + 9) \sqrt{5} = 13 \sqrt{(blank)}
\]

Next, we match the terms under the radicals. For the expression to match, \( \sqrt{(blank)} \) must also equal \( \sqrt{5} \), which means the blank is simply \( 5 \):

So we have:
\[
x + 9 = 13
\]

Now we can solve for \( x \):

\[
x = 13 - 9 = 4
\]

Thus, we can fill in the blanks:

\[
4 \sqrt{5} + 3 \sqrt{45} = 13 \sqrt{5}
\]

The complete expression is:

**4**√5 + 3 √45 = 13√**5**.

Answer 7

To solve for the blanks in the expression \( (blank)\sqrt{(blank)} - (blank)\sqrt{(blank)} = (blank)\sqrt{(blank)} \), we need to make sure the terms on the left side can be combined similarly to the radical on the right.

To provide a specific solution, let's use some values:

**Example:**
Let’s say we have:
- **First term:** \( 3\sqrt{8} \)
- **Second term:** \( 2\sqrt{8} \)

The expression will look like this:

\[
3\sqrt{8} - 2\sqrt{8} = (blank)\sqrt{(blank)}
\]

Now we can perform the subtraction:

\[
(3 - 2)\sqrt{8} = 1\sqrt{8}
\]

So, we have:

\[
1\sqrt{8}
\]

Now we can fill in the blanks:

- The first blank (multiplier of the first radical) is **3**.
- The second blank (the value under the first radical) is **8**.
- The third blank (multiplier of the second radical) is **2**.
- The fourth blank (the value under the second radical) is **8**.
- The fifth blank (multiplier of the result) is **1**.
- The sixth blank (the value under the resulting radical) is **8**.

Now putting it all together, our completed expression is:

\[
3\sqrt{8} - 2\sqrt{8} = 1\sqrt{8}
\]

Filling in the blanks:

**3**√**8** - **2**√**8** = **1**√**8**.

If you'd like a different set of numbers or a specific case, please let me know!

Answer 8

To find the product of the expression \( 8\sqrt{5} \cdot 3\sqrt{32} \), we will follow these steps:

1. **Multiply the coefficients**:
\[
8 \cdot 3 = 24
\]

2. **Multiply the radicals**:
\[
\sqrt{5} \cdot \sqrt{32} = \sqrt{5 \cdot 32} = \sqrt{160}
\]

Now, we handle \( \sqrt{160} \) by simplifying it:

3. **Factor \( 160 \)**:
\[
160 = 16 \cdot 10 = 16 \cdot 2 \cdot 5
\]
Therefore,
\[
\sqrt{160} = \sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10} = 4\sqrt{10}
\]

4. **Combine the results**:
Now we substitute back:
\[
8\sqrt{5} \cdot 3\sqrt{32} = 24\sqrt{160} = 24 \cdot 4\sqrt{10} = 96\sqrt{10}
\]

Now we can fill in the blanks:

**96**√**10**.

So the final answer is:

**96**√**10**.

Answer 9

To determine how many smaller right triangles with side lengths of \(5\sqrt{3}\) centimeters fit along the longer side of the larger right triangle with a side length of \(40\sqrt{12}\) centimeters, we need to divide the length of the larger triangle by the length of the smaller triangles.

1. **Simplify the length of the larger triangle**:
\[
40\sqrt{12} = 40\sqrt{4 \cdot 3} = 40 \cdot 2\sqrt{3} = 80\sqrt{3}
\]

2. **Calculate the number of smaller triangles**:
We find how many \(5\sqrt{3}\) lengths fit into \(80\sqrt{3}\):
\[
\text{Number of smaller triangles} = \frac{80\sqrt{3}}{5\sqrt{3}}
\]

The \(\sqrt{3}\) cancels out:
\[
= \frac{80}{5} = 16
\]

So, the number of smaller triangles that fit along the longer side is **16**.

Thus, the answer to fill in the blank is:

**16**.

Answer 10

To evaluate the expression \( \frac{47}{3} + \sqrt{121} \), we note that \( \sqrt{121} = 11 \), which is a rational number. Since the sum of two rational numbers (in this case, \( \frac{47}{3} \) and \( 11 \)) is also a rational number, the expression will produce a rational number.

Answer 11

To prove that the product of \( \sqrt{36} \) and \( \frac{5}{3} \) is a rational number, we first simplify \( \sqrt{36} \):

\[
\sqrt{36} = 6
\]

Next, we calculate the product:

\[
6 \cdot \frac{5}{3} = \frac{6 \cdot 5}{3} = \frac{30}{3} = 10
\]

The result \( 10 \) is a rational number since it can be expressed as the fraction \( \frac{10}{1} \). Additionally, the product of two rational numbers (in this case, \( 6 \) and \( \frac{5}{3} \)) is always a rational number, confirming that \( 10 \) is indeed rational.