State the restrictions on the variables and then simplify each of the following expressions.

A.) X root 12x^4
The answe is 2x^3 root 3 but I don't know how to get the answer it.
B.) (3x root x^2)(2 root x^5)
The answer is 6x^4 root X but I don't know how to get the answer.
Consider the equation -7= 2- root 12x-3. Solve the equation. The answer is X=7 but I can't figure out how to get the answer.

x√(12x^4) = x√(4x^4*3) = x(2x^2)√3 = 2x^3√3

3x√x^2 * 2√x^5 = 6x√(x^2*x^5) = 6x√(x^6 * x) = 6x*x^3 √x = 6x^4 √x

-7 = 2-√(12x-3)
-9 = -√(12x-3)
81 = 12x-3
84 = 12x
x = 7

A.) To simplify the expression X√(12x^4), we can start by simplifying the square root term. Since the square root of 12 is not a perfect square, we can factor it into its prime factors:

12 = 2 * 2 * 3

Next, we can simplify the expression by breaking down each factor with exponents.

X√(12x^4) = X√(2^2 * 3 * x^4)

Using the property of square roots ( √(ab) = √a * √b), we can break down the square root:

= X(√2^2 * √3 * √x^4)

Simplifying further, we can remove the square root of 2^2 (which is 2) since X is outside the square root:

= 2x^3√(3)

Therefore, the simplified expression is 2x^3√3.

B.) To simplify the expression (3x√x^2)(2√x^5), we can first simplify each term within the parentheses separately.

For the first term, we have 3x√(x^2). Since the square root of x^2 is x, we can simplify it to 3x * x = 3x^2.

For the second term, we have 2√(x^5). Since the square root of x^5 is x^(5/2), we can simplify it to 2 * x^(5/2).

Now, we can multiply the simplified terms:

(3x√x^2)(2√x^5) = 3x^2 * 2 * x^(5/2)

Simplifying further, we can multiply the coefficients and add the exponents of x:

= 6x^2 * x^(5/2) = 6x^((2+(5/2)))

To add the exponents, we need a common denominator:

= 6x^((4/2)+(5/2)) = 6x^(9/2)

Finally, we can simplify the exponent by dividing 9 by 2:

= 6x^(4.5)

Therefore, the simplified expression is 6x^(4.5)√x.

C.) Considering the equation -7 = 2 - √(12x - 3), we can solve it step-by-step.

Starting with the equation:
-7 = 2 - √(12x - 3)

First, let's isolate the square root term by subtracting 2 from both sides:
-7 - 2 = - √(12x - 3)

Simplifying:
-9 = - √(12x - 3)

Next, to get rid of the negative sign, we can square both sides of the equation:
(-9)^2 = (- √(12x - 3))^2

Simplifying further:
81 = 12x - 3

Next, let's isolate the variable by adding 3 to both sides:
81 + 3 = 12x - 3 + 3

Simplifying:
84 = 12x

Finally, divide both sides of the equation by 12 to solve for x:
84 / 12 = 12x / 12

Simplifying further:
7 = x

Therefore, the solution to the equation is x = 7.

A.) To simplify the expression "x√(12x^4)", we need to use the properties of radicals.

First, let's determine the restrictions on the variable "x" in this case. Since we have a square root (√) in the expression, the radicand (the value within the square root) must be non-negative. Therefore, the restriction is:
12x^4 ≥ 0

Simplifying 12x^4 ≥ 0:
Divide both sides by 12:
x^4 ≥ 0

Since x^4 will always be non-negative (any real number raised to an even power is non-negative), there are no restrictions on x.

Next, let's simplify the expression:

x√(12x^4)

Since the square root and the exponentiation have the same base (x), we can rewrite the expression as:
x^(1/2) * (12x^4)^(1/2)

Using the property of exponents, (ab)^c = a^c * b^c, we can simplify further:
x^(1/2) * (12^(1/2)) * (x^4)^(1/2)

The square root of 12 (√12) can be simplified as √(4 * 3) = √4 * √3 = 2√3.

So we have:
x^(1/2) * 2√3 * (x^4)^(1/2)

Now, using the property of exponents, (a^b)^c = a^(b*c), we can simplify further:
x^(1/2) * 2√3 * x^(4/2)

Multiply the coefficients:
2√3 * x^(1/2) * x^(4/2)

Since x^(1/2) * x^(4/2) = x^(1/2 + 4/2) = x^(5/2), we have:
2√3 * x^(5/2)

Finally, simplify further:
2x^(5/2)√3

This is the simplified expression for "x√(12x^4)". It is equal to 2x^(5/2)√3.

B.) To simplify the expression "(3x√(x^2))(2√(x^5))", we can use the same properties of radicals.

First, let's determine the restrictions on the variable "x" in this case. Since we have square roots (√) in the expression, the radicands (values within the square roots) must be non-negative. Therefore, the restrictions are:
x^2 ≥ 0 (from the first square root)
x^5 ≥ 0 (from the second square root)

Simplifying x^2 ≥ 0 and x^5 ≥ 0, we see that there are no specific restrictions on x.

Next, let's simplify the expression:

(3x√(x^2))(2√(x^5))

Using the property of exponents, x^(a*b) = (x^a)^b, and multiplying the coefficients, we have:
6x * √(x^(2+5))

Simplifying the exponents, we have:
6x * √(x^7)

This is the simplified expression for "(3x√(x^2))(2√(x^5))". It is equal to 6x√(x^7).

Now, let's move on to the equation:

-7 = 2 - √(12x - 3)

To solve this equation, we need to isolate the variable "x". First, let's move the constant term to the right side:

-7 + 7 = 2 - √(12x - 3) + 7

-7 cancels on the left side:
0 = 9 - √(12x - 3)

Now, let's isolate the radical term by subtracting 9 from both sides:

-9 = - √(12x - 3)

To eliminate the negative sign, we can square both sides of the equation:

(-9)^2 = (√(12x - 3))^2

81 = 12x - 3

Add 3 to both sides:

81 + 3 = 12x

84 = 12x

Divide both sides by 12:

84/12 = x

7 = x

Therefore, the solution to the equation -7 = 2 - √(12x - 3) is x = 7.