ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #471 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

44,000,000 can be written in scientific notation as \(\displaystyle a \times 10 ^{N}\) for some \(\displaystyle a, N\).

Which is the greater quantity?

(A) \(\displaystyle N\)

(B) 8

Possible Answers:

(B) is greater

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

Correct answer:

(B) is greater

Explanation:

To write 44,000,000 in scientifc notation, write the implied decimal point after the final "0", then move it left until it is after the first nonzero digit (the first "4").

\(\displaystyle 44 000 000. \Rightarrow 4.4 000 000\)

This requires a displacement of seven places, so  

\(\displaystyle 44,000,000 = 4.4 \times 10^{7}\)

\(\displaystyle N = 7 < 8\), and (B) is greater.

Example Question #11 : Exponential Operations

Give the result:

\(\displaystyle 333,335 ^{2} - 333,331 ^{2}\)

Possible Answers:

\(\displaystyle 2,000,004\)

\(\displaystyle 4,444,444\)

\(\displaystyle 4,000,004\)

\(\displaystyle 2,469,754\)

\(\displaystyle 2,666,664\)

Correct answer:

\(\displaystyle 2,666,664\)

Explanation:

For any two numbers \(\displaystyle a ,b\)

\(\displaystyle a^{2} - b ^{2} = (a+b) (a - b)\)

We can solve this problem most easily by taking advantage of this pattern, setting 

\(\displaystyle a = 333,335 , b =333,331\):

\(\displaystyle 333,335 ^{2} - 333,331 ^{2}\)

\(\displaystyle = \left (333,335 + 333,331 \right )\left (333,335 - 333,331 \right )\)

\(\displaystyle =666,666 \times 4\)

\(\displaystyle =2,666,664\)

Example Question #11 : Exponents

Raise \(\displaystyle 8 \times 10 ^{-3}\) to the fourth power and give the result in scientific notation.

Possible Answers:

\(\displaystyle 4.096 \times 10^{-84}\)

\(\displaystyle 3.2 \times 10^{-13}\)

\(\displaystyle 4.096 \times 10^{-9}\)

\(\displaystyle 3.2 \times 10^{-11}\)

\(\displaystyle 4.096 \times 10^{-15}\)

Correct answer:

\(\displaystyle 4.096 \times 10^{-9}\)

Explanation:

Use the properties of exponents to raise the number to the fourth power:

\(\displaystyle \left (8 \times 10 ^{-3} \right ) ^{4}\)

\(\displaystyle = 8 ^{4} \times \left ( 10 ^{-3} \right ) ^{4}\)

\(\displaystyle = 4,096 \times 10 ^{-3 \times 4 }\)

\(\displaystyle = 4,096 \times 10 ^{-12 }\)

This is not in scientific notation, so adjust:

\(\displaystyle 4.096 \times 10^{3} \times 10 ^{-12 }\)

\(\displaystyle = 4.096 \times 10^{3 + (-12)}\)

\(\displaystyle = 4.096 \times 10^{-9}\)

Example Question #111 : Numbers And Operations

Give the result:

\(\displaystyle 1,000,001 ^{2} - 999,999^{2}\)

Possible Answers:

\(\displaystyle 2,000,000\)

\(\displaystyle 1,000,000\)

\(\displaystyle 6,000,000\)

\(\displaystyle 8,000,000\)

\(\displaystyle 4,000,000\)

Correct answer:

\(\displaystyle 4,000,000\)

Explanation:

For any two numbers \(\displaystyle a ,b\)

\(\displaystyle a^{2} - b ^{2} = (a+b) (a - b)\)

We can solve this problem most easily by taking advantage of this pattern, setting 

\(\displaystyle a = 1,000,001 , b = 999,999\):

\(\displaystyle 1,000,001 ^{2} - 999,999^{2}\)

\(\displaystyle = \left ( 1,000,001+ 999,999 \right )\left ( 1,000,001- 999,999 \right )\)

\(\displaystyle = 2,000,000 \times 2 = 4,000,000\)

Example Question #112 : Numbers And Operations

Give the cube of \(\displaystyle 6 \times 10 ^{4}\)  in scientific notation.

Possible Answers:

\(\displaystyle 2.16 \times 10 ^{12 }\)

\(\displaystyle 1.8 \times 10 ^{12 }\)

\(\displaystyle 2.16 \times 10 ^{13}\)

\(\displaystyle 2.16 \times 10 ^{14 }\)

\(\displaystyle 1.8 \times 10 ^{13 }\)

Correct answer:

\(\displaystyle 2.16 \times 10 ^{14 }\)

Explanation:

\(\displaystyle \left (6 \times 10 ^{4} \right ) ^{3}\)

\(\displaystyle = 6^{3} \times \left (10 ^{4} \right ) ^{3}\)

\(\displaystyle =216 \times 10 ^{4 \times 3 }\)

\(\displaystyle =216 \times 10 ^{12 }\)

This is not in scientific notation, so adjust:

\(\displaystyle 2.16 \times 10 ^{2 } \times 10 ^{12 }\)

\(\displaystyle =2.16 \times 10 ^{2 + 12 }\)

\(\displaystyle =2.16 \times 10 ^{14 }\)

Example Question #472 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which expression is equal to 65,000?

Possible Answers:

\(\displaystyle 6.5*10^{4}\)

\(\displaystyle 6.5*10^{3}\)

\(\displaystyle 6.5*10^{5}\)

\(\displaystyle 65*10^{4}\)

Correct answer:

\(\displaystyle 6.5*10^{4}\)

Explanation:

\(\displaystyle 6.5*10^{4}\) is equal to \(\displaystyle 6.5*10,000\)

Move the decimal one place to the right for each number of the exponent with a base ten.

For example, \(\displaystyle 6.5 \ast10^{^{1}}=65\),  \(\displaystyle 6.5*10^{^{2}}=650\), etc.

 

Example Question #114 : Numbers And Operations

\(\displaystyle x^{3} = - 7\)

\(\displaystyle y ^{6} = 13\)

Evaluate \(\displaystyle (xy) ^{6}\).

Possible Answers:

\(\displaystyle 62\)

\(\displaystyle 637\)

\(\displaystyle -36\)

\(\displaystyle -637\)

Correct answer:

\(\displaystyle 637\)

Explanation:

By the Power of a Product Principle, 

\(\displaystyle (xy) ^{6} = x ^{6} \cdot y ^{6}\)

Also, by the Power of a Power Principle, 

\(\displaystyle (x^{3} ) ^{2} = x ^{3 \cdot 2 } = x ^{6}\)

Combining these ideas,

\(\displaystyle (xy) ^{6} = (x^{3} ) ^{2} \cdot y ^{6} = (-7 ) ^{2} \cdot 13 = 7 ^{2} \cdot 13 = 49 \cdot 13 = 637\)

Example Question #473 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

\(\displaystyle \frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}}\)

Possible Answers:

\(\displaystyle 3\)

\(\displaystyle 5\)

The expression is undefined.

\(\displaystyle 0\)

\(\displaystyle 1\)

Correct answer:

The expression is undefined.

Explanation:

\(\displaystyle \frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}} = \frac{ \left ( 30 -30\right )^{0}}{15^{0}}=\frac{ 0^{0}}{15^{0}}\)

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

Example Question #2 : How To Divide Exponents

Column A                  Column B

\(\displaystyle \left(\frac{1}{16}\right)^\frac{1}{2}\)                      \(\displaystyle \left(\frac{1}{16}\right)^{-\frac{1}{2}}\)

Possible Answers:

There is not enough information to determine the relationship between the quantities.

The quantity in Column B is greater.

The quantities are equal.

The quantity in Column A is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

Let's simplify both quantities first before we compare them. \(\displaystyle \left(\frac{1}{16}\right)^\frac{1}{2}\) becomes \(\displaystyle \sqrt{\frac{1}{16}}\)because the fractional exponent indicates a square root. We can simplify that by knowing that we can take the square roots of both the numerator and denominator, as shown by: \(\displaystyle \frac{\sqrt{1}}{\sqrt{16}}\). We can simplify further by taking the square roots (they're perfect squares) and get \(\displaystyle \frac{1}{4}\). Then, let's simplify Column B. To get rid of the negative exponent, we put the numerical expression on the denominator. There's still the fractional exponent at play, so we'll have a square root as well. It looks like this now: \(\displaystyle \frac{1}{\sqrt{\frac{1}{16}}}\). We already simplified \(\displaystyle \sqrt{\frac{1}{16}}\), so we can just plug in our answer, \(\displaystyle \frac{1}{4}\), into the denominator. Since we don't want a fraction in the denominator, we can multiply by the reciprocal of \(\displaystyle \frac{1}{4}\), which is 4 to get \(\displaystyle 1(4)\), which is just 4. Therefore, Column B is greater.

Example Question #2 : How To Divide Exponents

Give the reciprocal of \(\displaystyle 1.6 \times 10 ^{-7}\) in scientific notation.

Possible Answers:

\(\displaystyle 6.25 \times 10^{6 }\)

\(\displaystyle 6.25 \times 10^{7 }\)

\(\displaystyle 6.25 \times 10^{-7 }\)

\(\displaystyle 6.25 \times 10^{8 }\)

\(\displaystyle 6.25 \times 10^{-6 }\)

Correct answer:

\(\displaystyle 6.25 \times 10^{6 }\)

Explanation:

The reciprocal of \(\displaystyle 1.6 \times 10 ^{-7}\) is the quotient of 1 and the number;

\(\displaystyle \frac{1}{ 1.6 \times 10 ^{-7} }\)

\(\displaystyle = \frac{1 \times 10^{0}}{ 1.6 \times 10 ^{-7} }\)

\(\displaystyle = \frac{1 }{ 1.6 } \times \frac{ 10^{0}}{ 10 ^{-7} }\)

\(\displaystyle =0.625 \times 10^{0- (-7) }\)

\(\displaystyle =0.625 \times 10^{7 }\)

This is not in scientific notation, so adjust.

\(\displaystyle 6.25 \times 10^{-1 } \times 10^{7 }\)

\(\displaystyle = 6.25 \times 10^{-1+7 }\)

\(\displaystyle = 6.25 \times 10^{6 }\)

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