Explanation

Let's simplify the expression:

√3 × √5 × √15

= √(3 × 5 × 15)

= √(225)

= 15

Explanation

In the Cartesian coordinate system:

• 1st quadrant: (+, +)
• 2nd quadrant: (-, +)
• 3rd quadrant: (-, -)
• 4th quadrant: (+, -)

Given point P(-4, -7):

Both x (-4) and y (-7) coordinates are negative.

This point lies in the 3rd quadrant.

Explanation

Given:

a = 2

b = 3

c = 1

Let's calculate the expression:

a² + 3c + 4ab²

= (2)² + 3(1) + 4(2)(3)²

= 4 + 3 + 4(2)(9)

= 4 + 3 + 72

= 79

Explanation

Given the expression:

[(2/13)⁶ + (2/13)³]³ × (2/13)^-⁹

Let's simplify:

= [(2/13)³((2/13)³ + 1)]³ × (2/13)^-⁹

= (2/13)⁹[(2/13)³ + 1]³ × (2/13)^-⁹

The (2/13)⁹ and (2/13)^-⁹ cancel out:

= [(2/13)³ + 1]³

Given the result is 1, let's assume:

[(2/13)³ + 1]³ = 1

Explanation

1 + tan²θ = Sec²θ

This is a fundamental trigonometric identity.

Explanation

We are given:

• $x+y=6$

• $xy=5$

We are asked to find:
$x^2 + y^2$

Step-by-step solution:

Use the identity:

x^2+y^2=(x+y)^2−2xy

Substitute the given values:

$$$$x^2+y^2=(6)^2−2(5)=36−10=26

Explanation

Given y ∝ x, we can write y = kx.

When x = 5, y = 210:

210 = k(5)

k = 210/5

k = 42

Now, when x = 2:

y = kx

y = 42(2)

y = 84

Explanation

a² - b² = (a - b)(a + b)

a⁴ - b⁴ = (a² - b²)(a² + b²)

= (a - b)(a + b)(a² + b²)

The LCM is (a - b)(a + b)(a² + b²) = (a² - b²)(a² + b²)

= a⁴ - b⁴

Explanation

Given 12 = PQ, we can write P = 12/Q.

For proportion:

4 : Q = P : 3

4/ Q = P / 3

4 × 3 = P × Q

12 = PQ

This matches the given equation.

The answer is 4 : q = p : 3.

Explanation

Loga (m × n) = Loga m + Loga n.

This is a logarithmic property: the log of a product is the sum of the logs.