Answer :
Certainly! Let's break down each part of the given question step by step to find the values.
### Part (i)
The expression we need to solve in part (i) is:
[tex]\[ \sqrt{0.1849} + \sqrt{18 - 49} \][/tex]
First, we consider both terms separately.
- For \(\sqrt{0.1849}\), we recognize that 0.1849 is a positive number, so taking its square root is straightforward and real.
- Next, we look at \(\sqrt{18 - 49}\). The subtraction inside the square root gives us:
[tex]\[ 18 - 49 = -31 \][/tex]
The square root of a negative number is not defined within the real number system. Therefore,
[tex]\[ \sqrt{18 - 49} = \sqrt{-31} \quad \text{is undefined in the real numbers} \][/tex]
Since one of the terms is undefined, the entire expression \(\sqrt{0.1849} + \sqrt{18 - 49}\) is undefined in the real number system. Thus, the result for part (i) is:
[tex]\[ \text{undefined} \][/tex]
### Part (ii)
The expression we need to solve in part (ii) is:
[tex]\[ \sqrt{184900} - \sqrt{4 \times 18.49} \][/tex]
Let’s handle each square root separately.
1. First Term: \(\sqrt{184900}\)
- We are given that \(\sqrt{1849} = 43\). Since 184900 is simply 1849 multiplied by 100 (which implies shifting the decimal point), we can express this as:
[tex]\[ \sqrt{184900} = \sqrt{1849 \times 100} = \sqrt{1849} \times \sqrt{100} = 43 \times 10 = 430 \][/tex]
2. Second Term: \(\sqrt{4 \times 18.49}\)
- We first compute the value within the square root:
[tex]\[ 4 \times 18.49 = 73.96 \][/tex]
- Now we take the square root of 73.96:
[tex]\[ \sqrt{73.96} = 8.6 \][/tex]
Putting it all together, we subtract the second result from the first:
[tex]\[ \sqrt{184900} - \sqrt{4 \times 18.49} = 430 - 8.6 = 421.4 \][/tex]
Thus, the result for part (ii) is:
[tex]\[ 421.4 \][/tex]
### Summary
(i) \(\sqrt{0.1849} + \sqrt{18 - 49}\) is \text{undefined}.
(ii) [tex]\(\sqrt{184900} - \sqrt{4 \times 18.49}\)[/tex] is 421.4.
### Part (i)
The expression we need to solve in part (i) is:
[tex]\[ \sqrt{0.1849} + \sqrt{18 - 49} \][/tex]
First, we consider both terms separately.
- For \(\sqrt{0.1849}\), we recognize that 0.1849 is a positive number, so taking its square root is straightforward and real.
- Next, we look at \(\sqrt{18 - 49}\). The subtraction inside the square root gives us:
[tex]\[ 18 - 49 = -31 \][/tex]
The square root of a negative number is not defined within the real number system. Therefore,
[tex]\[ \sqrt{18 - 49} = \sqrt{-31} \quad \text{is undefined in the real numbers} \][/tex]
Since one of the terms is undefined, the entire expression \(\sqrt{0.1849} + \sqrt{18 - 49}\) is undefined in the real number system. Thus, the result for part (i) is:
[tex]\[ \text{undefined} \][/tex]
### Part (ii)
The expression we need to solve in part (ii) is:
[tex]\[ \sqrt{184900} - \sqrt{4 \times 18.49} \][/tex]
Let’s handle each square root separately.
1. First Term: \(\sqrt{184900}\)
- We are given that \(\sqrt{1849} = 43\). Since 184900 is simply 1849 multiplied by 100 (which implies shifting the decimal point), we can express this as:
[tex]\[ \sqrt{184900} = \sqrt{1849 \times 100} = \sqrt{1849} \times \sqrt{100} = 43 \times 10 = 430 \][/tex]
2. Second Term: \(\sqrt{4 \times 18.49}\)
- We first compute the value within the square root:
[tex]\[ 4 \times 18.49 = 73.96 \][/tex]
- Now we take the square root of 73.96:
[tex]\[ \sqrt{73.96} = 8.6 \][/tex]
Putting it all together, we subtract the second result from the first:
[tex]\[ \sqrt{184900} - \sqrt{4 \times 18.49} = 430 - 8.6 = 421.4 \][/tex]
Thus, the result for part (ii) is:
[tex]\[ 421.4 \][/tex]
### Summary
(i) \(\sqrt{0.1849} + \sqrt{18 - 49}\) is \text{undefined}.
(ii) [tex]\(\sqrt{184900} - \sqrt{4 \times 18.49}\)[/tex] is 421.4.
