Explanation

The volume of a parallelepiped is given by:

Volume = length × breadth × height

Substitute the given values:

Volume = 5x × 3x² × 7x⁴

Multiply the terms:

Volume = 105x⁷

Explanation

Given:

a + b = 5

a - b = √17

Square both equations:

(a + b)² = 5²

a² + 2ab + b² = 25

(a - b)² = (√17)²

a² - 2ab + b² = 17

Subtract the second equation from the first:

4ab = 8

Divide by 4:

ab = 2

Explanation

To find the percentage:

(72 ÷ 360) × 100% ≈ 20%

So, 72 is approximately 20% of 360.

Explanation

To make x² - xy + y² a perfect square, we need to add or subtract a term that will complete the square.

Recognize that:

x² - 2xy + y² = (x - y)²

However, the given expression is:

x² - xy + y²

To match the perfect square form, we need to subtract:

xy

So, subtracting xy will make it:

x² - 2xy + y² = (x - y)²

Explanation

Let x be the number of years in the future. Then:

Father’s age = 41 + x, Son’s age = 9 + x

Set up the equation: 41 + x = 3(9 + x)

→ 41 + x = 27 + 3x → 14 = 2x → x = 7 years.

Explanation

In the Cartesian plane, Quadrant II contains points where x is negative and y is positive.

Since (-3, 8) has x = -3 (negative) and y = 8 (positive), it lies in Quadrant II.

Explanation

A number is divisible by 8 if its last three digits form a number that is exactly divisible by 8.

For example, 4,320 is divisible by 8 because 320 ÷ 8 = 40 (a whole number).

Explanation

To factorize the equation:

a²b² - a² - b² + a

Rearrange the terms:

a²b² - b² - a² + a

Factor out common terms:

b²(a² - 1) - (a² - a)

Recognize the difference of squares:

b²(a + 1)(a - 1) - (a² - a)

Now, factor by grouping:

(a - 1)(ab² + b² - a)

Explanation

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data points.

It is calculated as the inverse of the average of the inverses of the values.

Explanation

This is a statement of the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side.

It ensures the sides form a valid triangle.