Question

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Factorise the following expressions -

1. a^4 - b^4
2. p^4 - 81
3. x^4 - (y+z)^4
4. x^4 - (x-z)^4

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1) A grandfather is ten tires older than his grandson. He is also 54 yers old then him. Their present age?

2) Gopi's age is 3 ties his son age. 10 years ago he was five time times his son age. Present age of both?

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Answer 1

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★ Question ★

★ Factorise the following expressions -

1) a^4 - b^4

2) p^4 - 81

3) x^4 - (y+z)^4

4) x^4 - (x-z)^4

★ Solution ★

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$1) \bf \: Given \: a {}^{4} - b {}^{4} \: = (a {}^{2} ) {}^{2} - (b {}^{2} ) {}^{2}$

$\bf = (a {}^{2} + b {}^{2} )(a {}^{2} - b {}^{2} )$

Formula :-

$\bf(x {}^{2} - {y}^{2} = (x + y)(x - y))$

Therefore answer

$\red{ = >} \bf \: ( {a}^{2} + {b}^{2} )(a + b)(a - b)$

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$\bf \: 2) \: given \: = {p}^{4} - 81$

$= \bf {p}^{4} - {3}^{4}$

$= \bf( {p}^{2} - {3}^{2} )( {p }^{2} + {3}^{2} )$

$\bf= (p - 3)(p + 3)( {p}^{2} - {3}^{2})$

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$\bf \: 3) Given \: = {x}^{4} - (y + z {}^{4} )$

$\bf = ( {x}^{2} ) {}^{2} - ({(y + z) {}^{2} }) {}^{2}$

$\bf = ( {x}^{2} + (y + z) {}^{2} )( {x}^{2} - (y + z) {}^{2} )$

$\bf = ( {x}^{2} + (y + z) {}^{2} )(x + y + z)(x - y - z)$

$\bf = ( {x}^{2} + {y}^{2} + {z}^{2} + 2yz)(x + y + z)(x - y - z)$

$\bf = 1 \times ( {x}^{2} + {y}^{2} + {z}^{2} ) \times (x + y + z) \times (x - y - z)$

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$\bf4) Given \: = {x}^{4} - (x - z) {}^{4}$

$\bf = ( {x}^{2} ) {}^{2} - ((x - z) {}^{2}) {}^{2}$

$\bf = ( {x}^{2} + (x - z) {}^{2} )( {x}^{2} - (x - z) {}^{2} ) {}^{2}$

$\bf= ( {x}^{2} + {x}^{2} - 2xz + {z}^{2} )(x + x - z)(x - x - z)$

$\bf = (2 {x}^{2} - 2xz + {z}^{2} )(2x - z)(z)$

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★ Question ★

A grandfather is ten tires older than his grandson. He is also 54 yers old then him. Their present age?

★ Solution ★

Let man’s present age to be x and his grandson’s be y

$\bf \: x=10y ...(1)$

$\bf \: x=y+54 ...(2)$

Put equation one in equation 2

$\bf \: 10y=y+54$

$\bf10y-y=54$

$\bf9y=54$

$\bf \: y= \frac{54}{9}$

$\bf \: Y=6$

$\bf \: X=10y$

$\bf \: X=10 \times 6$

$\bf \: X=60$

Man’s present age is 60 year and his grandson's present age is 6 years

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★ Question ★

Gopi's age is 3 ties his son age. 10 years ago he was five time times his son age. Present age of both?

★ Solution ★

$\bf \: Let \: Gopi's \: son \: age =x$

$\bf \: Gopi's \: age =3x$

Ten years ago, he was 5 times the age of his son.

$\bf \: Therefore, 3x−10=5(x−10)$

$\bf3x−10=5x−50$

$\bf2x=40$

$\bf \: x=20$

Gopi's son is 20 yrs old

Gopi is 3x=3×20=60 yrs old.

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★ Hope you understand this concept mate ★

Answer 2

${\large{\pmb{\sf{\underline{Required \; Qualitied \: Solution...}}}}}$

${\qquad \qquad \qquad{\pmb{\sf{\circ \: \: Question \: 1)}}}}$

➼ Factorise the following expressions -

• a⁴ - b⁴
• p⁴ - 81
• x⁴ - (y+z)⁴
• x⁴ - (x-z)⁴

➼ It is given that we have to factorise the given expressions. Let us factorise them properly by using suitable identity.

➼ Suitable identity is given below to solve all the following expressions -

${\small{\underline{\boxed{\sf{\bigstar \: \: (a^{2} - b^{2}) \: = (a+b)(a-b)}}}}}$

➼ Now let us solve all of them properly! Full Solution is given below!

⠀⠀⠀⠀⠀⠀Solution for part 1)

${\sf{:\implies a^{4} - b^{4}}}$

${\sf{:\implies (a^{2} - b^{2}) \: = (a+b)(a-b)}}$

${\sf{:\implies (a^{4} - b^{4}) \: = (a^{2})^{2} - (b^{2})^{2}}}$

${\sf{:\implies (a^{2} + b^{2}) (a^{2} - b^{2}}}$

${\sf{:\implies (a^{2} + b^{2}) (a+b) (a-b)}}$

⠀⠀⠀⠀⠀⠀Solution for part 2)

${\sf{:\implies p^{4} - 81}}$

${\sf{:\implies (a^{2} - b^{2}) \: = (a+b)(a-b)}}$

${\sf{:\implies p^{4} - 81 = (p^{2})^{2} - (9)^{2}}}$

${\sf{:\implies (p^{2} + 9) (p^{2} - 9)}}$

${\sf{:\implies (p^{2} + 9) (p^{2} - 3^{2})}}$

${\sf{:\implies (p^{2} + 9) (p+3) (p-3)}}$

⠀⠀⠀⠀⠀⠀Solution for part 3)

${\sf{:\implies x^{4} - (y+z)^{4}}}$

${\sf{:\implies (a^{2} - b^{2}) \: = (a+b)(a-b)}}$

${\sf{:\implies x^{4} - (y+z)^{4}}}$

${\sf{:\implies \therefore \: (x^{2})^{2} - [(y+z)^{2}]^{2}}}$

${\sf{:\implies [x^{2} + (y+z)^{2}] [x^{2}-(y+z)^{2}]}}$

${\sf{:\implies [x^{2} + y^{2} + z^{2} + 2yz] [(x+y+z) (x-y-z)]}}$

⠀⠀⠀⠀⠀⠀Solution for part 4)

${\sf{:\implies x^{4} - (x-z)^{4}}}$

${\sf{:\implies (a^{2} - b^{2}) \: = (a+b)(a-b)}}$

${\sf{:\implies x^{4} - (y+z)^{4}}}$

${\sf{:\implies \therefore \: (x^{2})^{2} - [(x-z)^{2}]}}$

${\sf{:\implies [x^{2} + (x-z)^{2}] . [x^{2} - (x-z)^{2}]}}$

${\sf{:\implies (x^{2} + x^{2} - 2xz + z^{2}) [(x+x-z) (x-x+z)]}}$

${\sf{:\implies (2x^{2} - 2xz + z^{2}) (2x-z) (z)}}$

${\qquad \qquad \qquad{\pmb{\sf{\circ \: \: Question \: 2)}}}}$

Given that: A grandfather is ten times older than his grand-son. He is also 54 years old than him.

⇢ Let the age of the grand-son be x

⇢ Let the age of grand-father be 10x

⠀⠀So according to the question,

${\sf{:\implies 10x = x+54}}$

${\sf{:\implies 10x - x = 54}}$

${\sf{:\implies 9x = 54}}$

${\sf{:\implies x = 54/9}}$

${\sf{:\implies x = 6 \: years}}$

Henceforth, the present age of the grandson is 6 years.

${\sf{:\implies 10x}}$

${\sf{:\implies 10(6)}}$

${\sf{:\implies 10 \times 6}}$

${\sf{:\implies 60 \: years}}$

Henceforth, 60 years is the present age of the grandfather.

${\qquad \qquad \qquad{\pmb{\sf{\circ \: \: Question \: 3)}}}}$

Given that: Gopi's age is 3 times her son age. 10 years ago she was five time times her son age.

⇢ Let the age of her son be x

⇢ Let the age of Gopi be 3x

⇢ Ten years ago, her son age be (x-10)

⇢ Ten years ago, her age be (3x-10)

⠀⠀So according to the question,

${\sf{:\implies 3x-10 = 5(x-10)}}$

${\sf{:\implies 3x-10 = 5x - 50}}$

${\sf{:\implies 3x-5x = -50+10}}$

${\sf{:\implies -2x = -40}}$

${\sf{:\implies 2x = 40}}$

${\sf{:\implies x = 40/2}}$

${\sf{:\implies x = 20}}$

Henceforth, 20 years is the present age of Gopi's son.

${\sf{:\implies 3x}}$

${\sf{:\implies 3(20)}}$

${\sf{:\implies 3 \times 20}}$

${\sf{:\implies 60 \: years}}$

Henceforth, 60 years is the present age of Gopi.