Answer:
Explanation:
Step 1: Let's start by simplifying -4x^2 * (√63x^2) using the following steps:
1.1: Separate √63x^2 into two terms:
-4x^2 * (√63 * √x^2)
1.2: Find the largest perfect square that we can factor out of 63. It's 9 as 3^2 = 9 and 9 * 7 = 63. Furthermore, √x^2 simplifies to x:
-4x^2 * (√9 * 7 * (x))
-4x^2 * (3√7 * x)
1.3: We can multiply x and 3:
-4x^2 * (3x√7)
1.4: We can distribute -4x^2 to 3x√7.
- Multiplying the coefficients will give us -12 as -4 * 3 = 12.
- x^2 * x = x^3
-12x^3 * √7
Step 2: Now we can work on simplifying x^3√112 using the following steps:
2.1: Find the largest perfect square you can factor out of 112. It's 16 since 4^2 = 16 and 16 * 7 = 112:
x^3 * √16 * 7
x^3 * 4√7
2.2: Multiply x^3 and 4√7:
4x^3 * √7
Step 3: Thus, -4x^2 * √63x^2 simplified individually is -12x^3 * √7 and x^3 * √112 simplified individually is 4x^3 * √7
3.1: Add -12x^3 * √7 + 4x^3 * √7:
-8x^3√7
Optional Step 4: We can check that we've correctly simplifed the equation by plugging in a number for x in both the radical expression not yet simplified and the simplified radical expression. If we get the same answer, then we've simplified the expression correctly. We can plug in 2 for x.
Plugging in 2 for x in -4x^2 * (√63x^2) + x^3 * (√112):
(1.) -4(2)^2 * √63(2)^2 + 2^3 * √112:
(2.) -4(4) * √63(4) + 8 * √112
(3.) -16 * √252 + 8 * √112
(4.) -16 * √36 * 7 + 8 *√16 * 7
(5.) -16 (6 * √7) + 8(4 *√7)
(6.) -96 * √7 + 32 * √7
(7.) -64√7
(8.) -169.3280839
Plugging in 2 for x in -8x^3√7:
(1.) -8(2)^3 * √7
(2.) -8(8) * √7
(3.) -64 * √7
(4.) -169.3280839
Therefore, we've correctly expressed the expression in simplest radical form