Exam 1: Review of Basic Concepts

Provide an appropriate response. -The formula used to find the volume of the frustum of a square pyramid is $V = \frac { 1 } { 3 } h \left( a ^ { 2 } + a b + b ^ { 2 } \right)$ , where $\mathrm { b }$ is the length of the base, $\mathrm { a }$ is the length of the top, and $\mathrm { h }$ is the height. Calculate the volume if $a = 271$ feet, $b = 671$ feet , and $h = 207$ feet.

Factor by any method. - $64 x ^ { 2 } - 64 y ^ { 2 }$

Simplify the expression. Assume all variables represent positive real numbers. - $- \sqrt { 54 }$

Provide an appropriate response. -The number of farms in a certain country since 1940, in millions, can be approximated by 308) evaluating the polynomial 0.000026045689x3 - 0.1522580959x2 + 296.5669762x - 192,464.713, where X is the year in question. Evaluate the polynomial for x = 2010 to determine the number of farms in 2010. Round the answer to the nearest million farms.

Find the product. Assume variables represent positive real numbers. - $6 x ^ { 6 / 5 } \left( x ^ { 11 / 5 } - 7 x \right)$

Factor, using the given factor. Assume all variables represent positive real numbers. - $6 ( 2 x - 1 ) ^ { - 3 / 4 } + 12 ( 2 x - 1 ) ^ { 1 / 4 } + 24 ( 2 x - 1 ) ^ { 5 / 4 } ; \quad 6 ( 2 x - 1 ) ^ { - 3 / 4 }$

Find the product. Assume variables represent positive real numbers. - $7 \mathrm { k } ^ { 1 / 8 } \left( 3 \mathrm { k } ^ { 3 / 8 } + 3 \mathrm { k } ^ { 3 / 4 } \right)$

Find the distance between two points given their coordinates. -Find the distance between points P and Q on a number line, with coordinates -4 and 2, respectively.

Write in radical form. Assume all variables represent positive real numbers. - $7 x ^ { 1 / 5 }$

Evaluate the expression. - $2401 ^ { 1 / 4 }$

Solve the problem. - $( 9 r - 10 ) ( 9 r + 10 )$

Factor, using the given factor. Assume all variables represent positive real numbers. - $( 4 m + 1 ) ^ { - 6 / 5 } + ( 4 m + 1 ) ^ { 1 / 5 } + ( 4 m + 1 ) ^ { 7 / 5 } ; \quad ( 4 m + 1 ) ^ { - 6 / 5 }$

Factor by any method. -Find all values of b that will make the polynomial a perfect square trinomial. $25 x ^ { 2 } + 40 x + b$

Solve the problem. - $\left( 4 s ^ { 2 } - 2 s + 4 \right) \left( s ^ { 2 } + 5 s - 2 \right)$

Simplify the expression. Assume all variables represent positive real numbers. - $\sqrt [ 3 ] { 8 k ^ { 15 } }$

Factor, using the given factor. Assume all variables represent positive real numbers. - $4 \mathrm { k } ^ { - 2 } + \mathrm { k } ^ { - 4 } ; \quad \mathrm { k } ^ { - 4 }$

Write the expression with only positive exponents and evaluate if possible. Assume all variables represent nonzero real numbers. - $- ( - 9 ) ^ { 0 }$

Use set notation, and list all the elements of the set. - $\left\{ 1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \ldots , \frac { 1 } { 1024 } \right\}$

Simplify the expression. Assume all variables represent positive real numbers. - $( 9 \sqrt { 3 } + 7 \sqrt { 2 } ) ( 4 \sqrt { 3 } + 9 \sqrt { 2 } )$

Factor the trinomial, if possible. - $- 30 x ^ { 2 } - 25 x + 30$