Sure, let's go through each question step-by-step.
Question 6: An operation is defined by [tex]\(a \square b = \sqrt{5a - 3b}\)[/tex]. State the value of [tex]\(4 \square 2\)[/tex].
To find the value of [tex]\(4 \square 2\)[/tex], we need to substitute [tex]\(a = 4\)[/tex] and [tex]\(b = 2\)[/tex] into the operation's formula.
The operation is given as:
[tex]\[ a \square b = \sqrt{5a - 3b} \][/tex]
Plugging in [tex]\(a = 4\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[ 4 \square 2 = \sqrt{5(4) - 3(2)} \][/tex]
Now, calculate inside the square root:
[tex]\[ 4 \square 2 = \sqrt{5 \cdot 4 - 3 \cdot 2} \][/tex]
[tex]\[ 4 \square 2 = \sqrt{20 - 6} \][/tex]
[tex]\[ 4 \square 2 = \sqrt{14} \][/tex]
Therefore, the value of [tex]\(4 \square 2\)[/tex] is approximately [tex]\(3.7416573867739413\)[/tex].
Question 7: An operation is defined by [tex]\(p \dagger q = 3p^2 - 2q\)[/tex]. Calculate the exact value of [tex]\(5 \dagger 3\)[/tex].
To find the value of [tex]\(5 \dagger 3\)[/tex], we need to substitute [tex]\(p = 5\)[/tex] and [tex]\(q = 3\)[/tex] into the operation's formula.
The operation is given as:
[tex]\[ p \dagger q = 3p^2 - 2q \][/tex]
Plugging in [tex]\(p = 5\)[/tex] and [tex]\(q = 3\)[/tex]:
[tex]\[ 5 \dagger 3 = 3(5)^2 - 2(3) \][/tex]
Now, calculate step-by-step:
[tex]\[ 5 \dagger 3 = 3 \cdot 25 - 2 \cdot 3 \][/tex]
[tex]\[ 5 \dagger 3 = 75 - 6 \][/tex]
[tex]\[ 5 \dagger 3 = 69 \][/tex]
Therefore, the exact value of [tex]\(5 \dagger 3\)[/tex] is [tex]\(69\)[/tex].
To summarize:
- The value of [tex]\(4 \square 2\)[/tex] is approximately [tex]\(3.7416573867739413\)[/tex].
- The value of [tex]\(5 \dagger 3\)[/tex] is exactly [tex]\(69\)[/tex].
