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ISEE Upper Level Quantitative : Variables

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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Example Question #11 : How To Find The Exponent Of Variables

is a negative number. Which is the greater quantity?

(a) $-16 a ^{4}$

(b) $(-2 a )^{4}$

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

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

Correct answer:

(b) is the greater quantity

Explanation:

Any nonzero number raised to an even power, such as 4, is a positive number. Therefore,

$-16 a ^{4} = -16 \cdot a ^{4}$ is the product of a negative number and a positive number, and is therefore negative.

By the same reasoning, $(-2 a )^{4}$ is a positive number.

It follows that $(-2 a )^{4} > -16 a ^{4}$ .

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Example Question #231 : Algebraic Concepts

$t ^{3} = -24$

Evaluate $\left ( \frac{2}{t} \right ) ^{6}$ .

Possible Answers:

$-\frac{1}{9}$

$\frac{1}{9}$

$-\frac{1}{72}$

$\frac{1}{72}$

Correct answer:

$\frac{1}{9}$

Explanation:

By the Power of a Power Principle,

$(t ^{3} )^{2} = t ^{3 \cdot 2 } = t ^{6}$

By way of the Power of a Quotient Principle,

$\left ( \frac{2}{t} \right ) ^{6} = \frac{2^{6}}{t^{6} } = \frac{2^{6}}{(t^{3}) ^{2}} = \frac{64}{576} = \frac{64 \div 64 }{576 \div 64} = \frac{1}{9}$ .

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Example Question #913 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are both real numbers.

$t^{4} = 121$

$u ^{4} = 81$

Evaluate (2t - 3u) (2t+3u ) .

Possible Answers:

Correct answer:

Explanation:

(2t - 3u) (2t+3u ) , as the product of a sum and a difference, can be rewritten using the difference of squares pattern:

(2t - 3u) (2t+3u )

$=(2t )^{2}-( 3u)^{2}$

$=2^{2} \cdot t ^{2}- 3^{2}\cdot u^{2}$

$=2^{2}t ^{2}- 3^{2}u^{2}$

$=4t ^{2}- 9u^{2}$

By the Power of a Power Principle,

$(t^{2} ) ^{2} = t^{2 \cdot 2 } = t^{4}$

Therefore, $t^{2}$ is a square root of $t^{4}$ - that is, a square root of 121. 121 has two square roots, and 121, but since is real, $t ^{2}$ must be the positive choice, 11. Similarly, $u^{2}$ is the positive square root of 81, which is 9.

The above expression can be evaluated as

$4t ^{2}- 9u^{2} = 4 (11) - 9 (9) = 44 - 81 = -37$ .

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Example Question #912 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

|t| = 4

|u| = 3

Which is the greater quantity?

(a) $(t- u) (t^{2}+ tu+u^{2})$

(b) 37

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

Correct answer:

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

Explanation:

Multiply the polynomials through distribution:

$(t- u) (t^{2}+ tu+u^{2})$

$= t (t^{2}+ tu+u^{2}) - u (t^{2}+ tu+u^{2})$

$= t \cdot t^{2}+ t \cdot tu+ t \cdot u^{2} - u \cdot t^{2}- u \cdot tu - u \cdot u^{2}$

$= t^{3}+ t^{2} u+ t u^{2} - t^{2} u - tu^{2} - u^{3}$

$= t^{3}+ t^{2} u- t^{2} u + t u^{2} - tu^{2} - u^{3}$

$= t^{3} - u^{3}$

The absolute value of is 4, so either t = -4 or t = 4 . Likewise, u = -3 or u = 3 .

If t = -4 and u = 3 , we see that

$(t- u) (t^{2}+ tu+u^{2})$

$= t^{3} - u^{3}$

$=(-4 )^{3} - 3^{3}$

= -64 - 27 = -91

If t = 4 and u = -3 , we see that

$(t- u) (t^{2}+ tu+u^{2})$

$= t^{3} - u^{3}$

$=4^{3} - (- 3)^{3}$

= 64 -( - 27 )= 91

In the first scenario, $(t- u) (t^{2}+ tu+u^{2}) < 37$ ; in the second, $(t- u) (t^{2}+ tu+u^{2}) > 37$ . This makes the information insufficient.

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Example Question #17 : How To Find The Exponent Of Variables

|t| = 4

|u| = 3

Which is the greater quantity?

(a) (4t+ 3u) (4t-3u)

(b) (3t+ 4u) (3t-4u)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

The absolute value of is 4, so either t = -4 or t = 4 . Likewise, u = -3 or u = 3 . However, since $(-4)^{2} = 4 ^{2} = 16$ and $(-3)^{2} = 3 ^{2} = 9$ , it follows that regardless, $t ^{2} = 16$ and $u^{2} = 9$ .

As the product of the sum and the difference of the same two expressions, (4t+ 3u) (4t-3u) can be rewritten as the difference of the squares of the expressions:

(4t+ 3u) (4t-3u)

$= (4t)^{2}-(3u)^{2}$

Using the Power of a Product Principle:

$(4t)^{2}-(3u)^{2}$

$= 4^{2}t^{2}-3^{2}u^{2}$

$=16t^{2}-9u^{2}$

Substituting,

$16t^{2}-9u^{2}$

$=16 \cdot 16 -9 \cdot 9$

= 256 - 81

= 175

Similarly,

(3t+ 4u) (3t-4u)

$= (3t)^{2}-(4u)^{2}$

$=3 ^{2}t^{2}-4^{2}u^{2}$

$=9t^{2}-16u^{2}$

$=9 \cdot 16 -16 \cdot 9$

= 144 - 144

= 0

Therefore, (4t+ 3u) (4t-3u) > (3t+ 4u) (3t-4u) .

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Example Question #21 : Variables And Exponents

$t ^{6}= 81$

$u ^{6}= 36$

Which is the greater quantity?

(a) $(t+ u) (t^{2}-tu+u^{2})$

(b) 16

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Multiply the polynomials through distribution:

$(t+ u) (t^{2}-tu+u^{2})$

$= t (t^{2}-tu+u^{2}) + u (t^{2}- tu+u^{2})$

$= t \cdot t^{2}- t \cdot tu+ t \cdot u^{2} + u \cdot t^{2}- u \cdot tu + u \cdot u^{2}$

$= t^{3}-t^{2} u+ t u^{2} +t^{2} u - tu^{2} +u^{3}$

Collecting like terms, the above becomes

$t^{3}+ t^{2} u- t^{2} u + t u^{2} - tu^{2} + u^{3}$

$= t^{3} + u^{3}$

By the Power of a Power Principle,

$\left (t^{3} \right ) ^{2} = t ^{3 \cdot 2} = t ^{6}$

This makes $t ^{3}$ a square root (positive or negative) of $t ^{6}$ , or 81, so

$t^{3} = - \sqrt{81} = -9$

$t^{3} = \sqrt{81} = 9$

We can not eliminate either since an odd power of a number can have any sign, and we are not given the sign of .

By similar reasoning, either

$u^{3} = - \sqrt{36} = -6$

$u^{3} = \sqrt{36} = 6$

$(t+ u) (t^{2}-tu+u^{2})$ can assume one of four values, depending on which values of $t^{3}$ and $u^{3}$ are selected:

$(t+ u) (t^{2}-tu+u^{2}) = t^{3} + u^{3} = 9 + 6 = 15$

$(t+ u) (t^{2}-tu+u^{2}) = t^{3} + u^{3} = 9 +( -6) = 3$

$(t+ u) (t^{2}-tu+u^{2}) = t^{3} + u^{3} = - 9 +6= -3$

$(t+ u) (t^{2}-tu+u^{2}) = t^{3} + u^{3} = - 9 +( -6) = -15$

Regardless of the choice of $t^{3}$ and $u^{3}$ , $16> (t+ u) (t^{2}-tu+u^{2})$ .

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Example Question #71 : Variables

Define $f(x) = x^{2}$ .

g (x) is a function with the set of all real numbers as its domain.

$(g \circ f)(5) = 60$

Which is the greater quantity?

(a) f(5)

(b) g(5)

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

Correct answer:

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

Explanation:

$f(x) = x^{2}$ , so $f(5) = 5^{2} = 25$ .

By definition,

$g[f(5)] = (g \circ f)(5)$ .

Since $(g \circ f)(5) = 60$ and f(5) = 25 , we can determine that

g (25 ) = 60 .

However, this does not tell us the value of g(x) at x = 5 . Therefore, we do not know whether f(5) or g(5) , if either, is the greater.

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Example Question #21 : Variables And Exponents

Simplify the expression:

$10x - 4x^{2} - 5x$

Possible Answers:

$-4x^{2} +15x$

$4x^{2} -5x$

$-4x^{2} +5x$

The expression cannot be simplified further.

$x^ {3}$

Correct answer:

$-4x^{2} +5x$

Explanation:

Group, then collect like terms.

$10x - 4x^{2} - 5x$

$= - 4x^{2} + 10x - 5x$

$= - 4x^{2} + \left ( 10 - 5 \right ) x$

$= - 4x^{2} + 5 x$

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Example Question #2 : How To Subtract Exponential Variables

Assume neither nor is zero.

Which is the greater quantity?

(a) $\frac{(x+y)^{2} - (x - y)^{2}}{xy}$

(b)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

Simplify the expression in (a):

$\frac{(x+y)^{2} - (x - y)^{2}}{xy}$

$=\frac{(x^{2}+2xy + y^{2}) - (x^{2}-2xy + y^{2})}{xy}$

$=\frac{x^{2}- x^{2}+2xy +2xy + y^{2}-y^{2}}{xy}$

$=\frac{4xy }{xy} = 4$

$\frac{(x+y)^{2} - (x - y)^{2}}{xy} = 4$ regardless of the values of the variables, and (a) and (b) are equal.

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Example Question #1 : How To Subtract Exponential Variables

is a positive number. is a negative number.

Which is the greater quantity?

(a) $x^{2} - y ^{2}$

(b) $(x-y) ^{2}$

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

$(x-y) ^{2} = x ^{2} - 2xy + y ^{2}$

is a positive number and is a negative number, so -2xy > 0 and $y^{2} > 0$ . Therefore,

$(x-y) ^{2} = x ^{2} - 2xy + y ^{2} > x ^{2} + y ^{2} > x ^{2} - y ^{2}$ .

(b) is always greater.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Variables

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : How To Find The Exponent Of Variables

Example Question #231 : Algebraic Concepts

Example Question #913 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #912 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #17 : How To Find The Exponent Of Variables

Example Question #21 : Variables And Exponents

Example Question #71 : Variables

Example Question #21 : Variables And Exponents

Example Question #2 : How To Subtract Exponential Variables

Example Question #1 : How To Subtract Exponential Variables

All ISEE Upper Level Quantitative Resources

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