simplify the the lowest term
1. square root of 40x^3
2. square root of 3/4 times square root of 4/7
3. square root of 6 + square root of 20
√(40x^3) = √(4x^2)√(10x) = 2x√(10x)
√(3/4) * √(4/7) = √(3*4 / 4*7) = √(3/7)
√6 + √20 = √6 + 2√5
To simplify the expressions and find the lowest term, we can follow a few steps for each one:
1. square root of 40x^3:
To simplify this expression, we can break down the number under the square root into its prime factors. In this case, the factors are:
40 = 2 * 2 * 2 * 5 = 2^3 * 5
Since the square root of x^2 is equal to x, we can rewrite the square root of 40x^3 as:
(square root of 40) * (square root of x^2 * x) = (square root of 40) * (x * square root of x) = x * 2 * square root of 10x.
Therefore, the simplified version of the expression is x * 2 * square root of 10x.
2. square root of 3/4 times square root of 4/7:
To simplify this expression, we can multiply the two square roots together. Recalling that the square root of a/b is equal to the square root of a divided by the square root of b:
(square root of 3/4) * (square root of 4/7) = (square root of 3/4) / (square root of 7).
To reduce the fraction under the square root, we can multiply both the numerator and the denominator by the square root of 4 to eliminate the square root in the denominator:
[(square root of 3) * (square root of 4)] / [(square root of 4) * (square root of 7)] = (square root of 12) / (2 * square root of 7).
Next, simplify the square root of 12 by breaking it down into its prime factors:
12 = 2 * 2 * 3 = 2^2 * 3
Substituting this into the expression gives us:
(sqrt(2^2 * 3)) / (2 * sqrt(7)) = (2 * sqrt(3)) / (2 * sqrt(7)).
Finally, we can simplify the expression by canceling out the 2's:
(sqrt(3)) / (sqrt(7)).
Therefore, the simplified version of the expression is sqrt(3) / sqrt(7).
3. square root of 6 + square root of 20:
To simplify this expression, we start by breaking down the numbers under the square roots into their prime factors:
6 = 2 * 3
20 = 2 * 2 * 5 = 2^2 * 5
Next, we can rewrite the expression as:
(sqrt(2 * 3)) + (sqrt(2^2 * 5)).
Now, simplify each square root separately:
(sqrt(2 * 3)) + (sqrt(2^2 * 5)) = sqrt(2) * sqrt(3) + 2 * sqrt(5).
Finally, combine like terms if possible, but in this case, the expression is already in its simplest form:
sqrt(2) * sqrt(3) + 2 * sqrt(5).
Therefore, there is no further simplification possible, and that is the lowest term.