Complete the expression adding the radicals.(1 point)

Question

(blank)√5 + 3 √45 = 13√(blank)

Answer 1

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 2

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

Answer 3

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 4

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

8√5–⋅3√32−−


(1 point)

(blank)√(blank)

Answer 5

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 6

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 7

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 8

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 9

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 10

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 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.