Exam 10: Rational Exponents, Radicals, and Complex Numbers

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 process of writing a radical expression as an equivalent expression but without a radical in the denominator is called ------the denominator.

Evaluate. -If $f ( x ) = \sqrt { 2 x - 6 }$ , find the value of $f ( 21 )$ .

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 find the area of the triangle if $a = 9 \mathrm {~cm} , \mathrm {~b} = 11 \mathrm {~cm}$ and $c = 16 \mathrm {~cm}$ .

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 9 } { \sqrt { 17 } + 4 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 7 } + \sqrt { 3 } ) ^ { 2 }$

Find the cube root. - $- \sqrt [ 3 ] { - 125 x ^ { 24 } y ^ { 18 } }$

Perform the indicated operation. Write the result in the form a + bi. -(3 + 9i) - (-2 + i)

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 50 } - 6 \sqrt { 162 } + 7 \sqrt { 128 }$

Use radical notation to write the expression. Simplify if possible. - $( - 64 ) ^ { 4 / 3 }$

Perform the indicated operation. Write the result in the form a + bi. -(8 - 3i) - (8 - i)

Evaluate. -If $f ( x ) = \sqrt { 2 x + 8 }$ , find the value of $f ( 7 )$ .

Use the properties of exponents to simplify the expression. Write with positive exponents. - $\left( - 8 p ^ { 3 / 5 } - 9 p ^ { 4 / 5 } \right) \left( - 8 p ^ { 3 / 5 } - 9 p ^ { 4 / 5 } \right)$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 4 ] { \frac { 16 } { 125 } }$

Solve the problem. -Police use the formula $s = \sqrt { 30 f d }$ to estimate the speed s of a car in miles per hour, where $\mathrm { d }$ is the distance in feet that the car skidded and $\mathrm { f }$ is the coefficient of friction. If the coefficient of friction on a certain dry road is $0.86$ and a car was traveling on it at a rate of $60 \mathrm { mph }$ , how far will the car skid? (Round to the nearest foot.)

Use rational exponents to simplify the following. - $\sqrt [ 21 ] { \mathrm { y } ^ { 14 } \mathrm { z } ^{18} }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 4 - 3 i } { 9 + 5 i }$

Write with positive exponents. Simplify if possible. - $\frac { 1 } { x ^ { - 7 / 8 } }$

Use the Pythagorean theorem to find the unknown side of the right triangle. -

Find the power of i. - $( - 2 i ) ^ { 5 }$

Find the cube root. - $\sqrt [ 3 ] { - 27 }$