Solve (sqrt{15}-sqrt{3})*(2sqrt{15}-4sqrt{3})

$$( \sqrt { 15 } - \sqrt { 3 } ) \cdot ( 2 \sqrt { 15 } - 4 \sqrt { 3 } )$$

Evaluate

$$42-18\sqrt{5}\approx 1.750776405$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{15}-\sqrt{3} by each term of 2\sqrt{15}-4\sqrt{3}.

$$2\left(\sqrt{15}\right)^{2}-4\sqrt{15}\sqrt{3}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{15} is 15.

$$2\times 15-4\sqrt{15}\sqrt{3}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

Multiply 2 and 15 to get 30.

$$30-4\sqrt{15}\sqrt{3}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$30-4\sqrt{3}\sqrt{5}\sqrt{3}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$30-4\times 3\sqrt{5}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

Multiply -4 and 3 to get -12.

$$30-12\sqrt{5}-2\sqrt{3}\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$30-12\sqrt{5}-2\sqrt{3}\sqrt{3}\sqrt{5}+4\left(\sqrt{3}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$30-12\sqrt{5}-2\times 3\sqrt{5}+4\left(\sqrt{3}\right)^{2}$$

Multiply -2 and 3 to get -6.

$$30-12\sqrt{5}-6\sqrt{5}+4\left(\sqrt{3}\right)^{2}$$

Combine -12\sqrt{5} and -6\sqrt{5} to get -18\sqrt{5}.

$$30-18\sqrt{5}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$30-18\sqrt{5}+4\times 3$$

Multiply 4 and 3 to get 12.

$$30-18\sqrt{5}+12$$

Add 30 and 12 to get 42.

$$42-18\sqrt{5}$$

Factor

$$6 {(7 - 3 \sqrt{5})} = 1.750776405$$