Exam 10: Rational Exponents, Radicals, and Complex Numbers

Solve. -If the three lengths of the sides of a triangle are known, Heron's formula can be used to find its area. and $c$ are the three lengths of the sides, Heron's formula for area is: $A = \sqrt { s ( s - a ) ( s - b ) ( s - c ) }$ where $s$ is half the perimeter of the triangle, or $s = \frac { 1 } { 2 } ( a + b + c )$ . Use this formula to approximate the area of the triangle to two decimal places, if necessary, when a $\mathrm { cm } , \mathrm { b } = 11 \mathrm {~cm}$ and $\mathrm { c } = 16 \mathrm {~cm}$ .

Evaluate. -If $f ( x ) = \sqrt [ 3 ] { x - 117 }$ , find the value of $f ( - 8 )$

Find the power of i. - $\mathrm { i } ^ { 28 }$

Find the root. Use absolute value bars when necessary. - $\sqrt [ 4 ] { ( 5 x z ) ^ { 4 } }$

Add or subtract. Assume all variables represent positive real numbers. - $7 \sqrt { 8 } - 10 \sqrt { 162 } - 3 \sqrt { 72 }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 135 a ^ { 4 } b ^ { 6 } } } { \sqrt [ 3 ] { 5 a } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 4 } { \sqrt [ 4 ] { a ^ { 3 } } }$

Perform the indicated operations. Assume that all variables represent positive numbers. - $\sqrt { 7 } ( \sqrt { 28 } + \sqrt { 14 } )$

Multiply or divide. - $\sqrt { 144 } \cdot \sqrt { - 1 }$

Find the distance between the pair of points. - $( - 5 , - 2 )$ and $( 7,2 )$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt [ 3 ] { 8 y } - \sqrt [ 3 ] { 54 y }$

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

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { 5 + \sqrt { 2 } } { 5 - \sqrt { 2 } }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 7 - 18 i } { - 6 i }$

Write with positive exponents. Simplify if possible. - $\frac { 3 } { 2 x ^ { - 7 / 6 } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 3 } { \sqrt { 125 x } }$

Find the square root. Assume that all variables represent positive real numbers. - $\sqrt { \frac { 49 } { 400 } }$

Fill in the blank with one of the words or phrases listed below. index rationalizing conjugate principal square rootcube root midpoint complex numberlike radicals radicand imaginary unit distance -The_____ formula is $\mathrm { d } = \sqrt { \left( \mathrm { x } _ { 2 } - \mathrm { x } _ { 1 } \right) ^ { 2 } + \left( \mathrm { y } _ { 2 } - \mathrm { y } _ { 1 } \right) ^ { 2 } }$ .

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $\sqrt { 2 } ( \sqrt { 14 } + \sqrt { 2 } )$

Write in terms of i. - $\sqrt { - 9 }$