Question

multiply and simplify:
$(5\sqrt{3} + 4)(\sqrt{21} - 3)$

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Explanation:

Step1: Apply the distributive property (FOIL method)

We use the formula \((a + b)(c - d)=ac - ad+bc - bd\). Here, \(a = 5\sqrt{3}\), \(b = 4\), \(c=\sqrt{21}\), \(d = 3\).
So, \((5\sqrt{3}+4)(\sqrt{21}-3)=5\sqrt{3}\times\sqrt{21}-5\sqrt{3}\times3 + 4\times\sqrt{21}-4\times3\)

Step2: Simplify each term

• For \(5\sqrt{3}\times\sqrt{21}\), we know that \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), so \(\sqrt{3}\times\sqrt{21}=\sqrt{3\times21}=\sqrt{63}=\sqrt{9\times7} = 3\sqrt{7}\). Then \(5\sqrt{3}\times\sqrt{21}=5\times3\sqrt{7}=15\sqrt{7}\)
• For \(-5\sqrt{3}\times3=-15\sqrt{3}\)
• For \(4\times\sqrt{21} = 4\sqrt{21}\)
• For \(-4\times3=-12\)

Step3: Combine like terms

There are no like terms to combine further. So the simplified form is \(15\sqrt{7}-15\sqrt{3}+4\sqrt{21}-12\)

Answer:

\(15\sqrt{7}-15\sqrt{3}+4\sqrt{21}-12\)