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Question Find the product of (27+36)(2 \sqrt{7}+3 \sqrt{6}) and (52+43)(5 \sqrt{2}+4 \sqrt{3}).

Studdy Solution

STEP 1

Assumptions
1. We are given two binomials of the form (ab+cd)(ef+gh)(a\sqrt{b} + c\sqrt{d})(e\sqrt{f} + g\sqrt{h}).
2. We will use the distributive property (FOIL method) to expand the product of these two binomials.
3. We will simplify the resulting terms by combining like terms and multiplying square roots.

STEP 2

First, we will apply the distributive property to multiply the two binomials.
(ab+cd)(ef+gh)=ab⋅ef+ab⋅gh+cd⋅ef+cd⋅gh(a\sqrt{b} + c\sqrt{d})(e\sqrt{f} + g\sqrt{h}) = a\sqrt{b} \cdot e\sqrt{f} + a\sqrt{b} \cdot g\sqrt{h} + c\sqrt{d} \cdot e\sqrt{f} + c\sqrt{d} \cdot g\sqrt{h}

STEP 3

Now, we will plug in the given values for a,b,c,d,e,f,g,a, b, c, d, e, f, g, and hh.
(27+36)(52+43)=27⋅52+27⋅43+36⋅52+36⋅43(2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3}) = 2\sqrt{7} \cdot 5\sqrt{2} + 2\sqrt{7} \cdot 4\sqrt{3} + 3\sqrt{6} \cdot 5\sqrt{2} + 3\sqrt{6} \cdot 4\sqrt{3}

STEP 4

Next, we will multiply the coefficients and the square roots separately.
27⋅52=107⋅22\sqrt{7} \cdot 5\sqrt{2} = 10\sqrt{7 \cdot 2} 27⋅43=87⋅32\sqrt{7} \cdot 4\sqrt{3} = 8\sqrt{7 \cdot 3} 36⋅52=156⋅23\sqrt{6} \cdot 5\sqrt{2} = 15\sqrt{6 \cdot 2} 36⋅43=126⋅33\sqrt{6} \cdot 4\sqrt{3} = 12\sqrt{6 \cdot 3}

STEP 5

Now we will simplify the square roots by multiplying the numbers under the square root.
107⋅2=101410\sqrt{7 \cdot 2} = 10\sqrt{14} 87⋅3=8218\sqrt{7 \cdot 3} = 8\sqrt{21} 156⋅2=151215\sqrt{6 \cdot 2} = 15\sqrt{12} 126⋅3=121812\sqrt{6 \cdot 3} = 12\sqrt{18}

STEP 6

We will further simplify the square roots by factoring out perfect squares.
1512=154⋅3=15⋅23=30315\sqrt{12} = 15\sqrt{4 \cdot 3} = 15 \cdot 2\sqrt{3} = 30\sqrt{3} 1218=129⋅2=12⋅32=36212\sqrt{18} = 12\sqrt{9 \cdot 2} = 12 \cdot 3\sqrt{2} = 36\sqrt{2}

STEP 7

Now we will combine the simplified terms.
1014+821+303+36210\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2}

STEP 8

We observe that there are no like terms to combine, so this is the final simplified form of the product.
The product is:
1014+821+303+36210\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2}
Therefore, the correct answer is:
1014+821+303+36210 \sqrt{14}+8 \sqrt{21}+30 \sqrt{3}+36 \sqrt{2}

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