Certainly! Let's solve each expression step-by-step.
### Expression 1: [tex]\( 3 \sqrt{1089} \)[/tex]
1. First, calculate the square root of 1089.
[tex]\( \sqrt{1089} = 33 \)[/tex]
2. Now, multiply the result by 3.
[tex]\( 3 \times 33 = 99 \)[/tex]
Thus, [tex]\( 3 \sqrt{1089} = 99 \)[/tex].
### Expression 2: [tex]\( 3 \sqrt{3^2 \times 11^2} \)[/tex]
1. Simplify inside the square root. Note that [tex]\( 3^2 \times 11^2 \)[/tex] is the same as [tex]\( (3 \times 11)^2 \)[/tex].
[tex]\( 3^2 \times 11^2 = (3 \times 11)^2 = 33^2 \)[/tex]
2. Now, take the square root of [tex]\( 33^2 \)[/tex].
[tex]\( \sqrt{33^2} = 33 \)[/tex]
3. Multiply the result by 3.
[tex]\( 3 \times 33 = 99 \)[/tex]
Thus, [tex]\( 3 \sqrt{3^2 \times 11^2} = 99 \)[/tex].
### Expression 3: [tex]\( 3 \sqrt{121} \)[/tex]
1. Calculate the square root of 121.
[tex]\( \sqrt{121} = 11 \)[/tex]
2. Now, multiply the result by 3.
[tex]\( 3 \times 11 = 33 \)[/tex]
Thus, [tex]\( 3 \sqrt{121} = 33 \)[/tex].
### Expression 4: [tex]\( 3 \sqrt{14^2} \)[/tex]
1. Calculate the square root of [tex]\( 14^2 \)[/tex].
[tex]\( \sqrt{14^2} = 14 \)[/tex]
2. Now, multiply the result by 3.
[tex]\( 3 \times 14 = 42 \)[/tex]
Thus, [tex]\( 3 \sqrt{14^2} = 42 \)[/tex].
### Expression 5: [tex]\( 3 \sqrt{11^3} \)[/tex]
1. Calculate [tex]\( 11^3 \)[/tex].
[tex]\( 11^3 = 1331 \)[/tex]
2. Calculate the square root of 1331. Note that [tex]\( 11^3 \)[/tex] can also be written as [tex]\( 11^{3/2} \)[/tex].
[tex]\( \sqrt{1331} = 36.4828726939094 \)[/tex]
4. Now, multiply the result by 3.
[tex]\( 3 \times 36.4828726939094 = 109.4486180817282 \)[/tex]
Thus, [tex]\( 3 \sqrt{11^3} \approx 109.4486180817282 \)[/tex].
To summarize, the solutions to the expressions are as follows:
1. [tex]\( 3 \sqrt{1089} = 99 \)[/tex]
2. [tex]\( 3 \sqrt{3^2 \times 11^2} = 99 \)[/tex]
3. [tex]\( 3 \sqrt{121} = 33 \)[/tex]
4. [tex]\( 3 \sqrt{14^2} = 42 \)[/tex]
5. [tex]\( 3 \sqrt{11^3} \approx 109.4486180817282 \)[/tex]
