Question

h. $3sqrt{18} + 3sqrt{12} + 4sqrt{27}$
j. $5sqrt{5} - 2sqrt{20}$

Explanation:

Part h: \( 3\sqrt{18} + 3\sqrt{12} + 4\sqrt{27} \)

Step1: Simplify each square root

Simplify \( \sqrt{18} \), \( \sqrt{12} \), and \( \sqrt{27} \) by factoring out perfect squares.
\( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \)
\( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
\( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \)

Step2: Substitute back into the expression

Substitute the simplified square roots into the original expression.
\( 3\sqrt{18} + 3\sqrt{12} + 4\sqrt{27} = 3(3\sqrt{2}) + 3(2\sqrt{3}) + 4(3\sqrt{3}) \)

Step3: Distribute the coefficients

Multiply the coefficients with the simplified square roots.
\( = 9\sqrt{2} + 6\sqrt{3} + 12\sqrt{3} \)

Step4: Combine like terms

Combine the terms with \( \sqrt{3} \).
\( = 9\sqrt{2} + (6\sqrt{3} + 12\sqrt{3}) \)
\( = 9\sqrt{2} + 18\sqrt{3} \)

Step1: Simplify \( \sqrt{20} \)

Factor out the perfect square from \( \sqrt{20} \).
\( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)

Step2: Substitute back into the expression

Substitute the simplified square root into the original expression.
\( 5\sqrt{5} - 2\sqrt{20} = 5\sqrt{5} - 2(2\sqrt{5}) \)

Step3: Distribute the coefficient

Multiply the coefficient with the simplified square root.
\( = 5\sqrt{5} - 4\sqrt{5} \)

Step4: Combine like terms

Subtract the terms with \( \sqrt{5} \).
\( = (5 - 4)\sqrt{5} \)
\( = \sqrt{5} \)

Answer:

\( 9\sqrt{2} + 18\sqrt{3} \)

Part j: \( 5\sqrt{5} - 2\sqrt{20} \) (assuming the '*' is a typo)