Exam 10: Rational Exponents, Radicals, and Complex Numbers

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 7 } }$

Write in terms of i. - $\sqrt { - 162 }$

Find the distance between the pair of points. - $( - 3,1 )$ and $( - 7,4 )$

Solve. - $\sqrt { 3 x + 5 } = 5$

Write the conjugate of the expression. - $\sqrt { 6 } + y$

Find the midpoint of the line segment whose endpoints are given. - $( - 9 , - 5 ) , ( 4,2 )$

Use radical notation to write the expression. Simplify if possible. - $16 ^ { 5 / 4 }$

Identify the domain and then graph the function. - $f(x)=\sqrt{x}+3$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 120 } } { \sqrt { 6 } }$

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive exponents. - $\left( \frac { 1 } { 9 } \right) ^ { 1 / 2 }$

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { 5 \sqrt { x } } { \sqrt { 7 y } }$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 5 x ^ { 2 } } + 7 \sqrt { 80 x ^ { 2 } } - 3 \sqrt { 80 x ^ { 2 } }$

Perform the indicated operation. Write the result in the form a + bi. -(3 + 7i) - (-2 + i)

Perform the indicated operation. Write the result in the form a + bi. -(5 - 2i) + (-5 + 2i)

Solve. - $\sqrt { x } - 1 = \sqrt { x - 7 }$

Use radical notation to write the expression. Simplify if possible. - $( 7 x ) ^ { 3 / 10 }$

Solve. - $x = \sqrt { 22 x - 55 } - 3$

Use the properties of exponents to simplify the expression. Write with positive exponents. - $\frac { y ^ { 3 / 4 } } { y ^ { 1 / 4 } }$

Rationalize the denominator. Assume that all variables represent positive numbers. - $\frac { \sqrt { x } - 4 } { \sqrt { x } + 4 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $2 \sqrt { 7 } ( \sqrt { 11 } + \sqrt { 7 } )$