Question

What is the product of the following expression in simplest radical form? sqrt(25) * sqrt(49)
Answer
What is the product of the following expression in simplest radical form? sqrt(25) * sqrt(49)
Answer
What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.

(8sqrt(27)) / (2sqrt(45))
Answered by GPT-4o mini
To simplify the expression \( \sqrt{25} \cdot \sqrt{49} \), we can first calculate each square root separately.

\[
\sqrt{25} = 5
\]
\[
\sqrt{49} = 7
\]

Now, we multiply these results together:

\[
\sqrt{25} \cdot \sqrt{49} = 5 \cdot 7 = 35
\]

Thus, the product of the expression in simplest radical form is

\[
\boxed{35}
\]
Answered by GPT-4o mini
To simplify the expression \( \sqrt{25} \cdot \sqrt{49} \), we can use the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

First, we compute:

\[
\sqrt{25} = 5
\]
\[
\sqrt{49} = 7
\]

Thus, we combine these:

\[
\sqrt{25} \cdot \sqrt{49} = 5 \cdot 7 = 35
\]

Alternatively, applying the property of square roots:

\[
\sqrt{25} \cdot \sqrt{49} = \sqrt{25 \cdot 49}
\]

Calculating \( 25 \cdot 49 \):

\[
25 \cdot 49 = 1225
\]

Then we find the square root:

\[
\sqrt{1225} = 35
\]

So, both methods confirm that the product of the expression in simplest radical form is

\[
\boxed{35}
\]
Answered by GPT-4o mini
To simplify the expression \( \frac{8\sqrt{27}}{2\sqrt{45}} \), we start by simplifying both the numerator and the denominator.

1. **Simplify the numerator:**
\[
8\sqrt{27} = 8\sqrt{9 \cdot 3} = 8\sqrt{9}\sqrt{3} = 8 \cdot 3 \cdot \sqrt{3} = 24\sqrt{3}
\]

2. **Simplify the denominator:**
\[
2\sqrt{45} = 2\sqrt{9 \cdot 5} = 2\sqrt{9}\sqrt{5} = 2 \cdot 3 \cdot \sqrt{5} = 6\sqrt{5}
\]

3. **Rewrite the expression using the simplified forms:**
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{24\sqrt{3}}{6\sqrt{5}}
\]

4. **Simplify the fraction:**
\[
\frac{24}{6} = 4
\]
Thus,
\[
\frac{24\sqrt{3}}{6\sqrt{5}} = 4\cdot\frac{\sqrt{3}}{\sqrt{5}}
\]

5. **Rationalize the denominator:**
To rationalize, we multiply the numerator and denominator by \(\sqrt{5}\):
\[
4 \cdot \frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{15}}{5} = \frac{4\sqrt{15}}{5}
\]

Thus, the quotient of the expression in simplest radical form, with the denominator rationalized, is

\[
\boxed{\frac{4\sqrt{15}}{5}}
\]