I can definitely help you with these math problems. Let's go through each question one by one and explain the steps to solve them.
1. What is the simplest form of the radical expression?
To simplify, let's consider each term separately:
3^3 sqrt(21) - 6^3 sqrt(2a)
First, let's simplify the terms within each square root:
sqrt(21) and sqrt(2a)
If there are no perfect square factors in the radicand, like in this case, we cannot simplify further.
Now, let's simplify the coefficients:
3^3 = 3 * 3 * 3 = 27
6^3 = 6 * 6 * 6 = 216
The expression becomes:
27sqrt(21) - 216sqrt(2a)
This is the simplest form of the radical expression.
2. Simplify.
When multiplying two radicals with the same index, we can combine them using the product rule:
3^(1/2) * 3^(1/2) = 3^(1/2 + 1/2) = 3^1 = 3
So, the simplified form of 3^(1/2) * 3^(1/2) is 3.
3. Rewrite ^n sqrt(x^m) using a rational expression.
To rewrite ^n sqrt(x^m) using a rational expression, we want to convert it into fractional exponent notation.
^n sqrt(x^m) = x^(m/n)
So, ^n sqrt(x^m) can be rewritten as x^(m/n).
4. Simplify.
sqrt(250x^16) / sqrt(2x)
To simplify this expression, we can combine the square roots by using the quotient rule:
sqrt(250x^16) / sqrt(2x) = sqrt(250x^16 / 2x)
Simplifying the fraction inside the square root gives:
sqrt(125x^15)
Now, let's simplify further:
sqrt(125x^15) = sqrt(25 * 5 * x^15)
Taking out perfect square factors, we have:
sqrt(25) * sqrt(5) * sqrt(x^15)
Simplifying:
5x^7 * sqrt(5)
So, the simplified form is 5x^7 sqrt(5).
5. What is the solution of the equation?
sqrt(2x + 13) - 5 = x
To solve this equation, let's isolate the square root term by adding 5 to both sides:
sqrt(2x + 13) = x + 5
Now, let's square both sides to eliminate the square root:
(2x + 13) = (x + 5)^2
Expanding the right side:
2x + 13 = x^2 + 10x + 25
Rearranging to form a quadratic equation:
x^2 + 8x + 12 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring gives:
(x + 2)(x + 6) = 0
Setting each factor equal to zero and solving for x:
x + 2 = 0 gives x = -2
x + 6 = 0 gives x = -6
So, the solutions to the equation are x = -2 and x = -6.
6. What is the solution of the equation?
3 ^5sqrt(x+2)^3 + 3 = 27
First, let's simplify the left side of the equation:
3 ^5sqrt(x+2)^3 = 3 * ((x+2)^(1/5))^3 = 3 * (x+2)^(3/5)
The equation becomes:
3 * (x+2)^(3/5) + 3 = 27
Next, let's isolate the term with the variable:
3 * (x + 2)^(3/5) = 27 - 3
Simplifying:
3 * (x + 2)^(3/5) = 24
Now, let's divide both sides by 3:
(x + 2)^(3/5) = 24 / 3
(x + 2)^(3/5) = 8
Taking the fifth power of both sides:
[(x + 2)^(3/5)]^5 = 8^5
(x + 2)^3 = 32768
Taking the cube root of both sides:
x + 2 = ^(3sqrt)32768
x + 2 = 32
Finally, solving for x:
x = 32 - 2
x = 30
So, the solution to the equation is x = 30.
I hope this explanation helps you understand how to solve these math problems. If you have any more questions, feel free to ask!