Let x=4x=4 and y=-2y=−2. Evaluate (x^2-y^2(10-y^2)-:3)^2(x2−y2(10−y2)÷3)2. Apparently I have to put a question mark here ?

Question from AoPS

INFO:
Subject: Prealgebra
Focus: Squares

1 Answer
MathFact-orials.blogspot.com
Jun 10, 2018

It reduces to 64

Explanation:

For questions of this type, we take the given values (x=4, y=-2)(x=4,y=−2) and substitute them into the expression to see what it simplifies to:

(x^2-y^2(10-y^2)-:3)^2(x2−y2(10−y2)÷3)2

(4^2-(-2)^2(10-(-2)^2)-:3)^2(42−(−2)2(10−(−2)2)÷3)2

Now that the values are placed, we now have to work through the order of operations:

  • color(red)(P)P - Parentheses (also known as Brackets)
  • color(blue)(E)E - Exponents
  • color(green)(M)M - Multiplication
  • color(green)(D)D - Division (this has the same weight as M and so I gave it the same colour)
  • color(brown)(A)A - Addition
  • color(brown)(S)S - Subtraction - (again, same weight as A and so the same colour)

The bracket containing the (10-(-2)^2)(10−(−2)2) term should be done first:

(10-(-2)^2)(10−(−2)2)

We first square the -2−2, then subtract that result from 10:

(10-4)=6(10−4)=6

:. (4^2-(-2)^2(6)-:3)^2

Let's now do the two squares remaining within the bracket:

(16-4(6)-:3)^2

Next we have the multiplication and division:

(16-24-:3)^2

(16-8)^2

We can now do the subtraction and then the square:

8^2=64

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