Numbers and Operations on Numbers

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Questions 1 - 10

In acres, give the area of a square plot of land three and one half miles in sidelength.

7,840 acres

3,920 acres

15,680 acres

4,480 acres

2,240 acres

Explanation

A square plot of land has as its area the square of the length of a side, so the land in question has area

$3 \frac{1}{2} \times 3 \frac{1}{2} = \frac{7}{2} \times \frac{7}{2} = \frac{49}{4} = 12\frac{1}{4}$ square miles.

One square mile is equal to 640 acres, so multiply this by the conversion factor 640:

$12\frac{1}{4} \textup{ mi}^{2} \times 640 \textup{ acres / }\textup{mi}^{2} = 7,840 \textup{ acres }$

Rewrite the sum of feet and 18 inches as a number of yards.

$\frac{2x+3}{6}$ yards

$\frac{3x+2}{6}$ yards

$\frac{2x+1}{6}$ yards

$\frac{x+2}{6}$ yards

$\frac{3x+1}{6}$

Explanation

One yard is equivalent to 3 feet, so feet can be converted to yards by dividing by the conversion factor 3:

$x \textup{ ft} \div 3\textup{ yd/ft} = \frac{x}{3} \textup{ yd}$

Also, one yard is equal to 36 inches, so 18 inches can be converted to yards by dividing by the conversion factor 36;

$18 \textup{ in} \div 36\textup{ in /yd} = \frac{18}{36} \textup{ yd} = \frac{1}{2} \textup{ yd}$

Add these quantities:

$\frac{x}{3} + \frac{1}{2} \textup{ yd}$

Rewriting with the least common denominator, which is 6:

$= \frac{x \times 2}{3\times 2} + \frac{1\times 3}{2\times 3} \textup{ yd}$

$= \frac{2x }{6} + \frac{ 3}{6} \textup{ yd}$

$= \frac{2x+3}{6} \textup{ yd}$ ,

the correct choice.

Rewrite the expression $\sqrt{x} \cdot \sqrt[3]{x}$ as a single radical expression.

$\sqrt[6]{x^{5}}$

$\sqrt[6]{x }$

$\sqrt[5]{x }$

$\sqrt[5]{x^{6} }$

$\sqrt[3]{x^{2}}$

Explanation

First, rewrite each radical expression using rational exponents:

$\sqrt{x} = x ^{\frac{1}{2}}$

$\sqrt[3]{x} = x ^{\frac{1}{3}}$

Multiply these expressions using the Product of Powers Property:

$\sqrt{x} \cdot \sqrt[3]{x}$

$= x ^{\frac{1}{2}} \cdot x ^{\frac{1}{3}}$

$= x ^{\frac{1}{2}+\frac{1}{3}}$

$= x ^{\frac{3}{6}+\frac{2}{6}}$

$= x ^{\frac{5}{6}}$

This can be rewritten in radical form as $\sqrt[6]{x^{5}}$ .

3 sticks of butter weigh 12 ounces combined. Using this information, determine the weight of 7 sticks of butter.

Explanation

Set up a proportional relationship, using to represent the unknown quantity (the weight of 7 sticks of butter):

$\frac{12\ ounces}{3\ sticks} = \frac{x\ ounces}{7\ sticks}$

Cross multiply:

$12\cdot 7 = 3\cdot x$

$84 = 3\cdot x$

Finally, divide both sides to solve for :

$\frac{84}{3} = x$

$x = 28\ ounces$

Simplify:

$\sqrt{120}$

$2\sqrt{30}$

The expression is already simplified.

$3\sqrt{20}$

$6\sqrt{10}$

$5\sqrt{12}$

Explanation

To simplify a radical expression, first find the prime factorization of the radicand, which is 120 here.

$120 = 2 \times 2 \times 2 \times 3 \times 5$

Pair up like factors, then apply the Product of Radicals Property:

$\sqrt{120} =\sqrt{ 2 \times 2 \times 2 \times 3 \times 5} =\sqrt{ 2 \times 2 } \times \sqrt{ 2 \times 3 \times 5} = 2 \sqrt{30}$ ,

the simplest form of the radical.

Simplify:

$\sqrt{32}$

$4\sqrt{2}$

The expression is already simplified.

$8\sqrt{2}$

$2\sqrt{8}$

$4\sqrt{8}$

Explanation

To simplify a radical expression, first find the prime factorization of the radicand, which is 32 here.

$32 = 2 \times 2 \times 2 \times 2 \times 2$

Pair up like factors, then apply the Product of Radicals Property:

$\sqrt{32} = \sqrt{2 \times 2} \times\sqrt{ 2 \times 2} \times \sqrt{2 }= 2 \times 2 \times \sqrt{2 } = 4 \sqrt{2 }$ ,

the simplest form of the radical.

Simplify the expression:

$\frac{10}{\sqrt{15}}$

$\frac{2\sqrt{15}}{3}$

$\sqrt{\frac{20}{3}}$

$\frac{\sqrt{60}}{3}$

$\frac{10\sqrt{15}}{15}$

$\sqrt{\frac{60}{9}}$

Explanation

An expression with a radical expression in the denominator is not simplified, so to simplify, it is necessary to rationalize the denominator. This is accomplished by multiplying both numerator and denominator by the given square root, $\sqrt{15}$ , as follows:

$\frac{10}{\sqrt{15}} = \frac{10 \times \sqrt{15}}{\sqrt{15} \times \sqrt{15}} = \frac{10 \sqrt{15}}{15}$

The expression can be simplified further by dividing the numbers outside the radical by greatest common factor 5:

$\frac{10 \sqrt{15}}{15} = \frac{(10 \div 5) \sqrt{15}}{15 \div 5} = \frac{2 \sqrt{15}}{3}$

This is the correct response.

Simplify:

$\sqrt{40}$

$4\sqrt{5}$

The expression is already simplified.

$2\sqrt{10}$

$5\sqrt{4}$

$5\sqrt{2}$

Explanation

To simplify a radical expression, first find the prime factorization of the radicand, which is 40 here.

$40 = 2 \times 2 \times 2 \times 5$

Pair up like factors, then apply the Product of Radicals Property:

$\sqrt{40 }= \sqrt{2 \times 2 \times 2 \times 5} = \sqrt{2 \times 2 } \times \sqrt{ 2 \times 5} = 2 \sqrt{10}$ ,

the simplest form of the radical.

How many significant digits does the number 0.03040 have?

Four

Two

Three

Five

Six

Explanation

In a decimal expression, any leading zeroes of a decimal expression must be discounted as significant digits; all digits from the first nonzero digit onward must be counted. These digits are in boldface below:

0.03040

Four is the correct response.

Simplify:

$\sqrt{32}$

$4\sqrt{2}$

The expression is already simplified.

$8\sqrt{2}$

$2\sqrt{8}$

$4\sqrt{8}$