Numbers and Operations on Numbers

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Questions 1 - 10

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 :

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

$x = 28\ ounces$

Give the missing term in the sequence:

$\left \{ 1 \right \}$

$\left \{ 1,2 \right \}$

$\left \{ 1,3 \right \}$

$\left \{ 1,2,4 \right \}$

$\left \{ 1,5\right \}$

$\left \{ 1,2, 3 , 6 \right \}$

$\left \{ 1,2,4, 8 \right \}$

$\left \{ 1,7 \right \}$

$\left \{ 1,2, 3, 4, 5, 6, 7 \right \}$

$\left \{ 1, 2 , 7 \right \}$

$\left \{ 1, 4 , 7 \right \}$

$\left \{ 7 \right \}$

Explanation

The sequence is one of sets. Note that in each term, the final element in each subsequent set increases by 1. Also note that each set comprises the set of all factors of its greatest element. Therefore, the missing set is the set that includes 7 and all of its other factors. Since 7 is a prime number, its set of factors is simply $\left \{ 1,7 \right \}$ , making this the correct response.

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.

Give the missing term in the sequence:

$\left \{ 1 \right \}$

$\left \{ 1,2 \right \}$

$\left \{ 1,3 \right \}$

$\left \{ 1,2,4 \right \}$

$\left \{ 1,5\right \}$

$\left \{ 1,2, 3 , 6 \right \}$

$\left \{ 1,2,4, 8 \right \}$

$\left \{ 1,7 \right \}$

$\left \{ 1,2, 3, 4, 5, 6, 7 \right \}$

$\left \{ 1, 2 , 7 \right \}$

$\left \{ 1, 4 , 7 \right \}$

$\left \{ 7 \right \}$

Explanation

Multiply, and express the product in scientific notation:

$7,200,000 \times 5,000,000$

$3.6 \times 10^{13}$

$3.6 \times 10^{12}$

$3.6 \times 10^{6}$

$3.6 \times 10^{36}$

$3.6 \times 10^{37}$

Explanation

Convert 7,200,000 to scientific notation as follows:

Move the (implied) decimal point until it is immediately after the first nonzero digit (the 7). This required moving the point six units to the left:

$7,200,000. \Rightarrow 7.200000$

It follows that 7,200,000 can be rewritten as $7.2 \times 10^{6}$ .

By similar reasoning, 5,000,000 can be rewritten as $5 \times 10^{6}$ .

Thus,

$7,200,000 \times 5,000,000$

$= (7.2\times 10 ^{6} )\times (5 \times 10^{6} )$

Rearrange and regroup the expressions so that the powers of ten are together:

$= (7.2\times 5) \times (10 ^{6} \times 10^{6} )$

Multiply the numbers in front. Also, multiply the powers of ten by adding exponents:

$= 36 \times 10 ^{6+6}$

$= 36 \times 10 ^{12}$

In order for the number to be in scientific notation, the number in front must be between 1 and 10. An adjustment must be made by moving the implied decimal point in 36 one unit left. It follows that

$36 = 3.6 \times 10^{1}$

Therefore,

$36 \times 10 ^{12} = (3.6 \times 10^{1}) \times 10 ^{12}$

$= 3.6 \times 10^{1+12}$

$= 3.6 \times 10^{13}$ ,

the correct response.

Simplify the following expression using scientific notation.

$(1.26\times10^6)+(2.4\times10^4)$

$1.284\times10^6$

$1.284\times10^5$

$1.2624\times10^6$

$1.5\times10^6$

$1.24677\times10^6$

Explanation

You can solve this problem in several ways. One way is to convert each number out of scientific notation and write it out fully, then find the sum of the two values and convert the answer back into scientific notation.

$1.26\times10^6+2.4\times10^4 \newline = 1,260,000+24,000 \newline = 1,284,000 \newline = 1.284\times10^6$

Another, potentially faster, way to solve this problem is to convert one answer into the same scientific-notational terms as the other and then sum them.

$1.26\times10^6+2.4\times10^4$

$2.4\times10^{4}=0.024\times10^{6}$

$1.26\times10^6+0.024\times10^6=(1.26+0.024)\times10^6=1.284\times10^6$

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

Rewrite as a single power of :

$(y^{2})^{3} \cdot y (y^{2})$

$y^{9}$

$y^{8}$

$y^{7}$

$y^{12}$

$y^{18}$

Explanation

Apply the properties of exponents to the expression as follows:

$(y^{2})^{3} \cdot y (y^{2})$

By the Power of a Power Property:

$= (y^{2 \cdot 3})\cdot y (y^{2})$

$= (y^{6})\cdot y (y^{2})$

Restating:

$= y^{6} \cdot y^{1} \cdot y^{2}$

By the Product of Powers of Property:

$= y^{6+1+2}$

$= y^{9}$ ,

the correct response.

Rewrite the sum of meters and 50 millimeters as a number of centimeters.

centimeters

$100x + \frac{1}{2}$ centimeters

Explanation

One meter is equivalent to 100 centimeters, so meters can be converted to centimeters by multiplying by conversion factor 100:

$x \textup{ m} \times 100 \textup{ cm/m} = 100x \textup{ cm}$

One centimeter is equivalent to 10 millimeters, so 50 millimeters can be converted to centimeters by dividing by conversion factor 10:

$50 \textup{ mm} \div 10 \textup{ mm/cm} = 5 \textup{ cm}$

Add these quantities:

$100x \textup{ cm} + 5 c= 100x + 5 \textup{ cm}$ ,

the correct response.

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.

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