Mathematics>Arithmetic>Cubes>Identities Based on Cube

EASY

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If $a + \frac{1}{a} = \sqrt{3}$ , find the value of $a^{3} + \frac{1}{a^{3}}$ .

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Important Questions on Cubes

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

Let

x = {(\sqrt{50} + 7)}^{\frac{1}{3}} - {(\sqrt{50} - 7)}^{\frac{1}{3}}

. Then-

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

The coefficient of

x

in

{(x - 3 y)}^{3}

is:

HARD

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a^{3} + \frac{1}{a^{3}} = 52

, then the value of

2 (a + \frac{1}{a})

is:

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

Let

a, b, c

be non-zero real numbers such that

a + b + c = 0;

let

q = a^{2} + b^{2} + c^{2} & r = a^{4} + b^{4} + c^{4}

. Then

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

x - y = 4

and

x y = 45

, then the value of

x^{3} - y^{3}

is:

HARD

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a^{4} + \frac{1}{a^{4}} = 50, a > 0,

then find the value of

a^{3} + \frac{1}{a^{3}}

.

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a + \frac{1}{a} = 5

then

a^{3} + \frac{1}{a^{3}}

is:

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

{(3 a - 4 b)}^{3}

is equal to:

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

p + q = 7

and

pq = 5

, then the value of

p^{3} + q^{3}

is:

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If P

= \frac{x^{4} - 8 x}{x^{3} - x^{2} - 2 x}

, Q

= \frac{x^{2} + 2 x + 1}{x^{2} - 4 x - 5}

and R

= \frac{2 x^{2} + 4 x + 8}{x - 5}

, then

(P \times Q) \div R

is equal to:

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a^{2} + b^{2} - c^{2} = 0

, then the value of

\frac{2 (a^{6} + b^{6} - c^{6})}{3 a^{2} b^{2} c^{2}}

is:

HARD

Mathematics>Arithmetic>Cubes>Identities Based on Cube

Consider the equation

{(1 + a + b)}^{2} = 3 (1 + a^{2} + b^{2}),

where

a, b

are real numbers. Then

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

Expand:

{(4 a + 3 b + 2 c)}^{2}

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a, b, c

are real numbers such that

a + b + c = 0

and

a^{2} + b^{2} + c^{2} = 1

, then

{(3 a + 5 b - 8 c)}^{2} + {(- 8 a + 3 b + 5 c)}^{2} + {(5 a - 8 b + 3 c)}^{2}

is equal to

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

A + B = 12

and

A B = 17

, what is the value of

A^{3} + B^{3}

?

EASY

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

{(2 x - 5 y)}^{3} - {(2 x + 5 y)}^{3} = y (A x^{2} + B y^{2})

, then what is the value of

(2 A - B)

?

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

a^{3} + b^{3} = 416

and

a + b = 16

, then

{(a - b)}^{2} + ab

is equals to_____.

HARD

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If

x^{2} - 2 \sqrt{5} x + 1 = 0

, then what is the value of

x^{5} + x^{\frac{1}{5}}

:

MEDIUM

Mathematics>Arithmetic>Cubes>Identities Based on Cube

If the difference of two numbers is

5

and the difference of their square is

125

, then out of the two which is the smaller number?

HARD

Mathematics>Arithmetic>Cubes>Identities Based on Cube

Let

x

and

y

be two

2

-digit numbers such that

y

is obtained by reversing the digits of

x .

Suppose they also satisfy

x^{2} - y^{2} = m^{2}

for some positive integer

m .

The value of

x + y + m

is.