Simplification...

Algebra Level 2

Evaluate 75 × 75 × 75 + 25 × 25 × 25 75 × 75 − 75 × 25 + 25 × 25 . \dfrac{{\color{#E81990}75 \times 75 \times 75} + {\color{#3D99F6}25 \times 25 \times 25}}{{\color{#E81990}75 \times 75} - {\color{#E81990}75} \times {\color{#3D99F6}25} + \color{#3D99F6}25 \times 25}.

Hint: Try using any one of the standard algebraic identities.

0 1 25 75 100

Ram Mohith by Ram Mohith , 18, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Ram Mohith
Jul 16, 2018

Let us assume that 75 = a , 25 = b {\color{#E81990}75 = a}, {\color{#3D99F6}25 = b}

Thus we can write the given expression as : a × a × a + b × b × b a × a − a × b + b × b = a 3 + b 3 a 2 − a b + b 2 \dfrac{a \times a \times a + b \times b \times b}{a \times a - a \times b + b \times b} = \dfrac{a^3 + b^3}{a^2 - ab + b^2}

But we know that a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3 + b^3 = (a + b)(a^2 - ab + b^2)

⟹ ( a + b ) ( a 2 − a b + b 2 ) ( a 2 − a b + b 2 ) \implies \dfrac{(a + b)\cancel{(a^2 - ab + b^2)}}{\cancel{(a^2 - ab + b^2)}}

⟹ a + b \implies {\color{#E81990}a} + {\color{#3D99F6}b}

⟹ 75 + 25 \implies {\color{#E81990}75} + {\color{#3D99F6}25}

⟹ 100 \implies {\color{#20A900}100}

6 Helpful 0 Interesting 1 Brilliant 0 Confused

It's a pretty obvious typo that is cleared up on the right side of the equation, but I think you meant "b + b x b" on the denominator of the fully extended version instead of "b + a x a."

Brian Bohan - 2 years, 7 months ago

Log in to reply

Thank you so much. I have edited it.

Ram Mohith - 2 years, 7 months ago
Chew-Seong Cheong
Jul 20, 2018

x = 75 × 75 × 75 + 25 × 25 × 25 75 × 75 − 75 × 25 + 25 × 25 Note that 75 = 3 × 25 = 3 × 25 × 3 × 3 + 25 3 × 3 − 3 + 1 Divide up and down by 25 × 25 = 25 ( 27 + 1 ) 7 = 25 × 4 = 100 \begin{aligned} x & = \frac {75\times 75\times 75+25\times 25\times 25}{75\times 75-75\times 25 + 25\times 25} & \small \color{#3D99F6} \text{Note that }75 = 3 \times 25 \\ & = \frac {3\times 25\times 3\times 3+25}{3\times 3-3 + 1} & \small \color{#3D99F6} \text{Divide up and down by }25 \times 25 \\ & = \frac {25(27+1)}7 \\ & = 25 \times 4 = \boxed{100} \end{aligned}

2 Helpful 0 Interesting 1 Brilliant 0 Confused

I took a somewhat similar approach but did not divide by 25 x 25 immediately. I took and decomposed 75 into 3 and 25 first to get 3^3 x 25 ^3 + 25^3 up top and 3^2 x 25^2 - 3 x 25^2 + 25^2. I then took the appropriate cubes and squares I need and got 27 x 25^3 + 25^3 on top and 9 x 25^2 - 3 x 25^2 + 25^2 on the bottom. This became 28 x 25^3 up top and 7 x 25^2 on the bottom. It was at this point that I divided 7 and 25^2 and got 4 x 25. :)

Brian Bohan - 2 years, 7 months ago
Jordan Cahn
Mar 19, 2019

Note that 25 = 3 × 25 {\color{magenta} 25} = 3\times {\color{#3D99F6} 25} . Then

75 × 75 × 75 + 25 × 25 × 25 75 × 75 − 75 × 25 + 25 × 25 = ( 3 × 25 ) 3 + 25 3 ( 3 × 25 ) 2 − 3 × 25 2 + 25 2 = 28 × 25 3 7 × 25 2 = 4 × 25 = 100 \begin{aligned} \frac{{\color{magenta} 75\times 75\times 75} + {\color{#3D99F6} 25\times 25\times 25}}{{\color{magenta} 75\times 75} - {\color{magenta} 75}\times{\color{#3D99F6} 25} + {\color{#3D99F6} 25\times 25}} &= \frac{(3\times {\color{#3D99F6}25})^3 + {\color{#3D99F6}25}^3}{(3\times {\color{#3D99F6}25})^2 - 3\times {\color{#3D99F6}25}^2 + {\color{#3D99F6}25}^2} \\ &= \frac{28\times {\color{#3D99F6}25}^3}{7\times {\color{#3D99F6}25}^2} \\ &= 4\times {\color{#3D99F6}25} \\ &= \boxed{100} \end{aligned}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...