Expand the following brackets:
\left(5 - \sqrt{13}\right) \left(5 + \sqrt{13}\right)
\left(\sqrt{11} - 11\right) \left(\sqrt{11} + 11\right)
\left( 8 \sqrt{5} - 6\right) \left( 8 \sqrt{5} + 6\right)
\left(\sqrt{7} + \sqrt{5}\right) \left(\sqrt{7} - \sqrt{5}\right)
\left( 7 \sqrt{11} - \sqrt{7}\right) \left( 7 \sqrt{11} + \sqrt{7}\right)
\left( 5 \sqrt{3} + 3 \sqrt{5}\right) \left( 5 \sqrt{3} - 3 \sqrt{5}\right)
Expand the following brackets:
\left(\sqrt{3} - 13\right)^{2}
\left(\sqrt{7} + \sqrt{3}\right)^{2}
\left( 3 \sqrt{3} + 8\right)^{2}
\left( 4 \sqrt{2} - \sqrt{13}\right)^{2}
\left( 3 \sqrt{2} + 4 \sqrt{13}\right)^{2}
\left(5 \sqrt{2} - \sqrt{32} \right)^{2}
Expand the following brackets:
\left(\sqrt{11} + 10\right) \left(\sqrt{3} - 9\right)
\left(\sqrt{11} - \sqrt{13}\right) \left(\sqrt{7} - \sqrt{2}\right)
\left( 11 \sqrt{2} - \sqrt{7}\right) \left( 13 \sqrt{3} - \sqrt{5}\right)
\left( 4 \sqrt{2} - \sqrt{7}\right) \left( 3 \sqrt{3} + \sqrt{8}\right)
\left( 17 \sqrt{3} - 8\sqrt{8}\right) \left(\sqrt{24} - \sqrt{5}\right)
\left( \sqrt{90} - 7\sqrt{7}\right) \left(\sqrt{72} - 6\sqrt{5}\right)
Consider the following equation:
\left( 3 \sqrt{35} - 2 \sqrt{7}\right)^{2} = x - y \sqrt{5}
Expand and simplify the left hand side of the equation.
State the values of x and y.
Consider the following equation:
\left(\sqrt{m} + n\right)^{2} = 16 + 6 \sqrt{7}
Expand and simplify the left hand side of the equation.
State the values of m and n.
Consider the following expression:
\left(\sqrt{a} + 7\right)^{2} = 54+b \sqrt{5}
Expand and simplify the left hand side of the equation.
State the values of a and b.
Consider the following expression:
\left(x\sqrt{y} + 5\right)^{2} = 33+ 4\sqrt{2}
Expand and simplify the left habd side of the equation.
State the values of x and y.
Consider the following expression:
\left(\sqrt{7} + 4\right)^{2} + \left(\sqrt{7} + m\right)^{2}
Expand and simplify the expression.
Hence what value of m can be substituted into \left(\sqrt{7} + 4\right)^{2} + \left(\sqrt{7} + m\right)^{2} so that it has a rational value?
Consider the following right triangle:
Find c, the length of the hypotenuse of the triangle.
Find the exact perimeter of the figure.
Find the exact area of the figure.
Consider the following isosceles triangle:
Find h, the height of the isosceles triangle.
Find the exact perimeter of the triangle.
Find the exact area of the triangle.
Define rational and irrational numbers and perform operations with surds and fractional indices.