Simply:
(a) Four multiply by square root of seven open bracket two multiply by the square root of three plus three multiply by the square root of six minus the square root of seven close bracket.
(b) open bracket three multiply by the square root of seven minus the square root of two close bracket squared.
(c) sixteen multiply by the square root of twenty four divided by four multiply by the square root of ninety six.
(d) Seven divide by the square root of twenty minus four divided by square root twelve.
(e) Three multiply by the square root of two minus five multiply by the square root of three divided by the square root of eight.
(a) To simplify the expression:
Four multiplied by the square root of seven times (2 multiplied by the square root of three) plus (3 multiplied by the square root of six) minus the square root of seven.
Let's break it down step-by-step:
Step 1: Simplify inside the brackets.
2 multiplied by the square root of three becomes 2√3.
Step 2: Distribute the Four to both terms inside the brackets.
Four multiplied by the square root of seven times 2√3 becomes 8√3√7.
Step 3: Simplify the radical expressions.
The product of two radicals simplifies to a single radical, so 8√3√7 becomes 8√21.
Step 4: Evaluate the remaining terms.
The final expression is 8√21 plus (3 multiplied by the square root of six) minus the square root of seven.
(b) To simplify the expression:
(3 multiplied by the square root of seven) minus the square root of two, all squared.
Let's break it down step-by-step:
Step 1: Simplify inside the brackets.
3 multiplied by the square root of seven remains the same.
Step 2: Square the expression.
(3 multiplied by the square root of seven) squared becomes (3√7)^2.
Step 3: Evaluate the square.
(3√7)^2 simplifies to 9 * 7, which is 63.
The final expression is 63.
(c) To simplify the expression:
16 multiplied by the square root of twenty-four, divided by (4 multiplied by the square root of ninety-six).
Let's break it down step-by-step:
Step 1: Simplify inside the brackets.
4 multiplied by the square root of ninety-six becomes 4√96.
Step 2: Simplify the radical expressions.
The square root of twenty-four can be simplified to 2√6, and the square root of ninety-six can be simplified to 4√6.
16 multiplied by 2√6 divided by (4 multiplied by 4√6).
Step 3: Cancel out common factors.
Cancel out the common factor of 4 in the numerator and denominator.
4 divided by 1 remains the same.
The final expression is 4.
(d) To simplify the expression:
7 divided by the square root of twenty minus 4 divided by the square root of twelve.
Let's break it down step-by-step:
Step 1: Simplify the radicals.
The square root of twenty can be simplified to 2√5, and the square root of twelve can be simplified to 2√3.
7 divided by 2√5 minus 4 divided by 2√3.
Step 2: Evaluate each term separately.
Divide 7 by 2√5 and 4 by 2√3.
The final expression is (7/2√5) - (4/2√3).
(e) To simplify the expression:
3 multiplied by the square root of two minus 5 multiplied by the square root of three, divided by the square root of eight.
Let's break it down step-by-step:
Step 1: Simplify the radicals.
The square root of two, square root of three, and square root of eight cannot be simplified further.
3√2 - 5√3 divided by √8.
Step 2: Simplify the denominator.
The square root of eight can be simplified to 2√2.
3√2 - 5√3 divided by 2√2.
Step 3: Rationalize the denominator.
Multiply both the numerator and denominator by the conjugate of the denominator (√2).
(3√2 - 5√3) * (√2) divided by (2√2) * (√2).
Simplifying further, we get:
(3√4 - 5√6) divided by (2√4).
Step 4: Simplify the numerator and denominator.
The square root of 4 is 2.
(3*2 - 5√6) divided by (2*2).
The final expression is (6 - 5√6) divided by 4.
To simplify these expressions, we can follow a few general rules:
1. Simplify any square roots where possible.
2. Perform any multiplication or division operations.
3. Simplify further if necessary.
(a) Four multiplied by the square root of seven, multiplied by (two multiplied by the square root of three plus three multiplied by the square root of six, minus the square root of seven.)
To simplify this expression, we need to distribute the 4 to both terms inside the parentheses:
4 * 2 * sqrt(7) + 4 * 3 * sqrt(6) - 4 * sqrt(7)
This gives us:
8 * sqrt(7) + 12 * sqrt(6) - 4 * sqrt(7)
Since both the first and last term contain sqrt(7), we can combine them:
(8 - 4) * sqrt(7) + 12 * sqrt(6)
4 * sqrt(7) + 12 * sqrt(6)
This is the simplified expression.
(b) (3 multiplied by the square root of seven, minus the square root of two) squared.
To simplify this expression, we square the entire expression inside the parentheses:
(3 * sqrt(7) - sqrt(2))^2
Using the formula (a - b)^2 = a^2 - 2ab + b^2, we expand and simplify:
9 * 7 - 2 * 3 * sqrt(7) * sqrt(2) + 2
63 - 6 * sqrt(14) + 2
65 - 6 * sqrt(14)
This is the simplified expression.
(c) Sixteen multiplied by the square root of twenty-four, divided by four multiplied by the square root of ninety-six.
First, simplify the square roots:
sqrt(24) = sqrt(4 * 6) = 2 * sqrt(6)
sqrt(96) = sqrt(16 * 6) = 4 * sqrt(6)
Now, substitute these values back into the expression:
16 * 2 * sqrt(6) / 4 * 4 * sqrt(6)
Cancel out the common factors:
(16 * 2) / (4 * 4)
32 / 16
2
This is the simplified expression.
(d) Seven divided by the square root of twenty, minus four divided by square root twelve.
First, simplify the square roots:
sqrt(20) = sqrt(4 * 5) = 2 * sqrt(5)
sqrt(12) = sqrt(4 * 3) = 2 * sqrt(3)
Now, substitute these values back into the expression:
7 / 2 * sqrt(5) - 4 / 2 * sqrt(3)
Simplify further:
7/2 * sqrt(5) - 2 * sqrt(3)
This is the simplified expression.
(e) Three multiplied by the square root of two, minus five multiplied by the square root of three, divided by the square root of eight.
First, simplify the square root:
sqrt(8) = sqrt(4 * 2) = 2 * sqrt(2)
Now, substitute this value back into the expression:
3 * sqrt(2) - 5 * sqrt(3) / 2 * sqrt(2)
Simplify further by canceling out the common factors:
(3 - 5 * sqrt(3)) / 2
This is the simplified expression.