Question
1224√3300√
Answers
GPT-4o mini
To evaluate \( 1224 \sqrt{3300} \), we first simplify \( \sqrt{3300} \).
1. Factor \( 3300 \):
\[
3300 = 33 \times 100 = 33 \times 10^2
\]
\[
33 = 3 \times 11 \quad \text{and} \quad 100 = 10^2 = (10)^2
\]
Thus, we can write:
\[
3300 = 3 \times 11 \times 100 = 3 \times 11 \times 10^2
\]
2. Taking the square root:
\[
\sqrt{3300} = \sqrt{3 \times 11 \times 100} = \sqrt{3 \times 11} \times \sqrt{100} = 10 \sqrt{33}
\]
3. Substitute back into the expression:
\[
1224 \sqrt{3300} = 1224 \times 10 \sqrt{33}
\]
\[
= 12240 \sqrt{33}
\]
Thus, the value of \( 1224 \sqrt{3300} \) simplifies to:
\[
\boxed{12240 \sqrt{33}}
\]
1. Factor \( 3300 \):
\[
3300 = 33 \times 100 = 33 \times 10^2
\]
\[
33 = 3 \times 11 \quad \text{and} \quad 100 = 10^2 = (10)^2
\]
Thus, we can write:
\[
3300 = 3 \times 11 \times 100 = 3 \times 11 \times 10^2
\]
2. Taking the square root:
\[
\sqrt{3300} = \sqrt{3 \times 11 \times 100} = \sqrt{3 \times 11} \times \sqrt{100} = 10 \sqrt{33}
\]
3. Substitute back into the expression:
\[
1224 \sqrt{3300} = 1224 \times 10 \sqrt{33}
\]
\[
= 12240 \sqrt{33}
\]
Thus, the value of \( 1224 \sqrt{3300} \) simplifies to:
\[
\boxed{12240 \sqrt{33}}
\]
12/sqrt24 divided by 3/sqrt300
GPT-4o mini
To solve the expression \( \frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}} \), we can first rewrite the division as a multiplication by the reciprocal:
\[
\frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}} = \frac{12}{\sqrt{24}} \times \frac{\sqrt{300}}{3}
\]
Now, we can multiply the fractions:
\[
= \frac{12 \cdot \sqrt{300}}{3 \cdot \sqrt{24}}
\]
Next, simplify the fraction:
\[
= \frac{12}{3} \cdot \frac{\sqrt{300}}{\sqrt{24}} = 4 \cdot \frac{\sqrt{300}}{\sqrt{24}} = 4 \cdot \sqrt{\frac{300}{24}}
\]
Now, simplify the fraction inside the square root:
\[
\frac{300}{24} = \frac{300 \div 12}{24 \div 12} = \frac{25}{2}
\]
So, we have:
\[
\sqrt{\frac{300}{24}} = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}}
\]
Now substituting back, we have:
\[
4 \cdot \frac{5}{\sqrt{2}} = \frac{20}{\sqrt{2}}
\]
For final simplification, we can rationalize the denominator:
\[
\frac{20}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2}
\]
Thus, the final result is:
\[
\boxed{10\sqrt{2}}
\]
\[
\frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}} = \frac{12}{\sqrt{24}} \times \frac{\sqrt{300}}{3}
\]
Now, we can multiply the fractions:
\[
= \frac{12 \cdot \sqrt{300}}{3 \cdot \sqrt{24}}
\]
Next, simplify the fraction:
\[
= \frac{12}{3} \cdot \frac{\sqrt{300}}{\sqrt{24}} = 4 \cdot \frac{\sqrt{300}}{\sqrt{24}} = 4 \cdot \sqrt{\frac{300}{24}}
\]
Now, simplify the fraction inside the square root:
\[
\frac{300}{24} = \frac{300 \div 12}{24 \div 12} = \frac{25}{2}
\]
So, we have:
\[
\sqrt{\frac{300}{24}} = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}}
\]
Now substituting back, we have:
\[
4 \cdot \frac{5}{\sqrt{2}} = \frac{20}{\sqrt{2}}
\]
For final simplification, we can rationalize the denominator:
\[
\frac{20}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2}
\]
Thus, the final result is:
\[
\boxed{10\sqrt{2}}
\]
What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.
12/24√3/300√
12/24√3/300√
GPT-4o mini
To calculate the quotient of the expression \(\frac{12}{24\sqrt{3}} \div \frac{300}{\sqrt{}} \), we will first clarify the expression as follows:
\[
\frac{12}{24\sqrt{3}} \div \frac{300}{\sqrt{}} \equiv \frac{12}{24\sqrt{3}} \div \frac{300}{1} = \frac{12}{24\sqrt{3}} \times \frac{1}{300}
\]
We can rewrite this as:
\[
\frac{12 \cdot 1}{24\sqrt{3} \cdot 300}
\]
Now, we can simplify the numerator and denominator:
1. **Simplify \( \frac{12}{24} \)**:
\[
\frac{12}{24} = \frac{1}{2}
\]
Now, we substitute this back into the expression:
\[
= \frac{1}{2\sqrt{3} \cdot 300} = \frac{1}{600\sqrt{3}}
\]
2. **Rationalize the denominator**:
To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{3}\):
\[
\frac{1 \cdot \sqrt{3}}{600\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{600 \cdot 3} = \frac{\sqrt{3}}{1800}
\]
Thus, the final answer in simplest radical form is:
\[
\boxed{\frac{\sqrt{3}}{1800}}
\]
\[
\frac{12}{24\sqrt{3}} \div \frac{300}{\sqrt{}} \equiv \frac{12}{24\sqrt{3}} \div \frac{300}{1} = \frac{12}{24\sqrt{3}} \times \frac{1}{300}
\]
We can rewrite this as:
\[
\frac{12 \cdot 1}{24\sqrt{3} \cdot 300}
\]
Now, we can simplify the numerator and denominator:
1. **Simplify \( \frac{12}{24} \)**:
\[
\frac{12}{24} = \frac{1}{2}
\]
Now, we substitute this back into the expression:
\[
= \frac{1}{2\sqrt{3} \cdot 300} = \frac{1}{600\sqrt{3}}
\]
2. **Rationalize the denominator**:
To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{3}\):
\[
\frac{1 \cdot \sqrt{3}}{600\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{600 \cdot 3} = \frac{\sqrt{3}}{1800}
\]
Thus, the final answer in simplest radical form is:
\[
\boxed{\frac{\sqrt{3}}{1800}}
\]
What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.
12/√24/3/√300
12/√24/3/√300
GPT-4o mini
To find the quotient of the expression \(\frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}}\), we first rewrite the division as multiplication by the reciprocal:
\[
\frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}} = \frac{12}{\sqrt{24}} \times \frac{\sqrt{300}}{3}
\]
Now we can combine the two fractions:
\[
= \frac{12 \cdot \sqrt{300}}{3 \cdot \sqrt{24}}
\]
Next, we simplify:
1. **Reduce \( \frac{12}{3} \)**:
\[
\frac{12}{3} = 4
\]
So our expression becomes:
\[
= \frac{4\sqrt{300}}{\sqrt{24}}
\]
Now, let's simplify \(\frac{\sqrt{300}}{\sqrt{24}}\):
\[
\frac{\sqrt{300}}{\sqrt{24}} = \sqrt{\frac{300}{24}}
\]
Now we simplify \(\frac{300}{24}\):
\[
\frac{300}{24} = \frac{300 \div 12}{24 \div 12} = \frac{25}{2}
\]
So we have:
\[
\sqrt{\frac{300}{24}} = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}}
\]
Now substituting back, we get:
\[
= 4 \cdot \frac{5}{\sqrt{2}} = \frac{20}{\sqrt{2}}
\]
Next, we rationalize the denominator:
\[
\frac{20}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2}
\]
Thus, the final answer in simplest radical form is:
\[
\boxed{10\sqrt{2}}
\]
\[
\frac{12}{\sqrt{24}} \div \frac{3}{\sqrt{300}} = \frac{12}{\sqrt{24}} \times \frac{\sqrt{300}}{3}
\]
Now we can combine the two fractions:
\[
= \frac{12 \cdot \sqrt{300}}{3 \cdot \sqrt{24}}
\]
Next, we simplify:
1. **Reduce \( \frac{12}{3} \)**:
\[
\frac{12}{3} = 4
\]
So our expression becomes:
\[
= \frac{4\sqrt{300}}{\sqrt{24}}
\]
Now, let's simplify \(\frac{\sqrt{300}}{\sqrt{24}}\):
\[
\frac{\sqrt{300}}{\sqrt{24}} = \sqrt{\frac{300}{24}}
\]
Now we simplify \(\frac{300}{24}\):
\[
\frac{300}{24} = \frac{300 \div 12}{24 \div 12} = \frac{25}{2}
\]
So we have:
\[
\sqrt{\frac{300}{24}} = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}}
\]
Now substituting back, we get:
\[
= 4 \cdot \frac{5}{\sqrt{2}} = \frac{20}{\sqrt{2}}
\]
Next, we rationalize the denominator:
\[
\frac{20}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2}
\]
Thus, the final answer in simplest radical form is:
\[
\boxed{10\sqrt{2}}
\]