What is the simplified form of (6x^-2)^2(0.5x)^4 expressed without negative exponents? Can you verify my solution?
To simplify the expression (6x^-2)^2(0.5x)^4 without negative exponents, we can simplify each term separately and then multiply them together.
Let's start with the first term, (6x^-2)^2. To simplify this, we need to apply the exponent of 2 to both the base (6) and the exponent (-2).
First, let's square the base 6: 6^2 = 36.
Next, we need to square the exponent -2. When we square a negative exponent, it becomes positive. So, (-2)^2 = 4.
Combining these results, we have (6x^-2)^2 = 36x^4.
Now let's simplify the second term, (0.5x)^4. To do this, we need to apply the exponent of 4 to both the base (0.5) and the exponent (x).
First, let's raise the base 0.5 to the power of 4: (0.5)^4 = 0.0625.
Next, we need to raise the exponent x to the power of 4: x^4.
Combining these results, we have (0.5x)^4 = 0.0625x^4.
Finally, to express the simplified form without negative exponents, we can multiply the two simplified terms together:
36x^4 * 0.0625x^4 = 2.25x^8.
So, the simplified form of (6x^-2)^2(0.5x)^4 without negative exponents is 2.25x^8.
To verify if your solution is correct, you can compare it with the one we derived.
To find the simplified form of the expression (6x^-2)^2(0.5x)^4 without negative exponents, we can follow these steps:
Step 1: Simplify the first term (6x^-2)^2:
(6x^-2)^2 = 6^2 * (x^-2)^2
= 36 * x^(-2 * 2)
= 36 * x^-4
Step 2: Simplify the second term (0.5x)^4:
(0.5x)^4 = 0.5^4 * (x)^4
= 0.0625 * x^4
Step 3: Multiply the simplified terms from step 1 and step 2:
(36 * x^-4) * (0.0625 * x^4)
= 36 * 0.0625 * x^-4 * x^4
= 2.25 * x^(-4 + 4)
= 2.25 * x^0
= 2.25
Therefore, the simplified form of (6x^-2)^2(0.5x)^4 expressed without negative exponents is 2.25.
To verify this solution, we can substitute a value for x and check if the original expression and the simplified form yield the same result. Let's say x = 3:
Original expression = (6(3)^-2)^2(0.5(3))^4
= (6/9)^2(1.5)^4
= (2/3)^2(1.5)^4
= (4/9)(2.25)^4
= (4/9)(2.25 * 2.25 * 2.25 * 2.25)
≈ 10.417
Simplified form = 2.25
We can see that the values are not the same, so the simplified form is incorrect. Please double-check your calculations.