6. An operation is defined by [tex]a \square b = \sqrt{5a - 3b}[/tex]. State the value of [tex]4 \square 2[/tex].

7. An operation is defined by [tex]p \dagger q = 3p^2 - 2q[/tex]. Calculate the exact value of [tex]5 \dagger 3[/tex].



Answer :

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].