(a+sqrt(7a))*(2a+sqrt(4a))

Apply Distributive Property: To multiply the two expressions, we will use the distributive property (also known as the FOIL method for binomials), which states that for any numbers x, y, z, and w, (x + y)(z + w) = xz + xw + yz + yw.Multiply First Terms: First, we multiply the first terms of each binomial: a * 2a = 2a^2.Multiply Outer Terms: Next, we multiply the outer terms: a * sqrt(4a) = a * 2sqrt(a) = 2a^(3/2).Multiply Inner Terms: Then, we multiply the inner terms: sqrt(7a) * 2a = 2a^(3/2) * sqrt(7) = 2sqrt(7) * a^(3/2).Multiply Last Terms: Finally, we multiply the last terms of each binomial: sqrt(7a) * sqrt(4a) = sqrt(28a^2) = sqrt(4*7*a^2) = 2a * sqrt(7) because sqrt(4) = 2 and sqrt(a^2) = a.Add All Products: Now, we add all the products together: 2a^2 + 2a^(3/2) + 2sqrt(7) * a^(3/2) + 2a * sqrt(7).Combine Like Terms: We can combine like terms: 2a^2 + (2 + 2sqrt(7)) * a^(3/2).

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(a+\sqrt{7 a}) \cdot(2 a+\sqrt{4 a})

Apply Distributive Property: To multiply the two expressions, we will use the distributive property (also known as the FOIL method for binomials), which states that for any numbers $x$ , $y$ , $z$ , and $w$ , $(x + y)(z + w) = xz + xw + yz + yw$ .
Multiply First Terms: First, we multiply the first terms of each binomial: $a \times 2a = 2a^2$ .
Multiply Outer Terms: Next, we multiply the outer terms: $a \times \sqrt{4a} = a \times 2\sqrt{a} = 2a^{\frac{3}{2}}$ .
Multiply Inner Terms: Then, we multiply the inner terms: $\sqrt{7a} \times 2a = 2a^{\frac{3}{2}} \times \sqrt{7} = 2\sqrt{7} \times a^{\frac{3}{2}}$ .
Multiply Last Terms: Finally, we multiply the last terms of each binomial: $\sqrt{7a} \times \sqrt{4a} = \sqrt{28a^2} = \sqrt{4\times7\times a^2} = 2a \times \sqrt{7}$ because $\sqrt{4} = 2$ and $\sqrt{a^2} = a$ .
Add All Products: Now, we add all the products together: $2a^2 + 2a^{\frac{3}{2}} + 2\sqrt{7} \cdot a^{\frac{3}{2}} + 2a \cdot \sqrt{7}$ .
Combine Like Terms: We can combine like terms: $2a^2 + (2 + 2\sqrt{7}) \cdot a^{\frac{3}{2}}$ .