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ISEE Upper Level Quantitative : Variables and Exponents

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #21 : Variables And Exponents

|t| = 4

|u| = 3

Which is the greater quantity?

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

(b) 37

Possible Answers:

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

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 #22 : Variables And Exponents

|t| = 4

|u| = 3

Which is the greater quantity?

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

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

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

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 #23 : 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

(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:

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 #24 : Variables And Exponents

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

(a) and (b) are equal

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

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:

$4x^{2} -5x$

$x^ {3}$

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

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

The expression cannot be simplified further.

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 #22 : Variables And Exponents

Assume neither nor is zero.

Which is the greater quantity?

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

(b)

Possible Answers:

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

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 #23 : Variables And Exponents

is a positive number. is a negative number.

Which is the greater quantity?

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

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

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) 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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Example Question #1 : How To Multiply Exponential Variables

Simplify:

$\left (4x \right )^{4} +\left ( 2y \right )^{2}$

Possible Answers:

$64x^{4} y ^{2}$

$4x^{4} +2y ^{2}$

$1,024x^{4} y ^{2}$

$16x^{4} +4y ^{2}$

$256x^{4} +4y ^{2}$

Correct answer:

$256x^{4} +4y ^{2}$

Explanation:

$\left (4x \right )^{4} +\left ( 2y \right )^{2} =4^{4}x^{4} + 2^{2}y^{2}=256x^{4} +4y ^{2}$

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Example Question #2 : How To Multiply Exponential Variables

Expand: $(x-5)^{3}$

Possible Answers:

$x^{3}+15x^{2}-75x-125$

$x^{3}-25x^{2}+50x-125$

$x^{3}-125$

$x^{3}+25x^{2}-50x-125$

$x^{3}-15x^{2}+75x-125$

Correct answer:

$x^{3}-15x^{2}+75x-125$

Explanation:

A binomial can be cubed using the pattern:

$(A-B)^{3} = A^{3} - 3A^{2}B+ 3AB^{2} - B^{3}$

Set A = x, B = 5

$(x-5)^{3}$

$= x^{3} - 3x^{2} \cdot 5+ 3x\cdot 5^{2} - 5^{3}$

$= x^{3} - 15x^{2} + 75x - 125$

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Example Question #3 : How To Multiply Exponential Variables

Factor completely:

$3x^{2} - x -14$

Possible Answers:

$\left ( x+2 \right ) \left (3x- 7 \right )$

$\left ( x+7 \right ) \left (3x-2 \right )$

$\left ( x-7 \right ) \left (3x+2 \right )$

$\left ( x-2 \right ) \left (3x- 7 \right )$

$\left ( x-2 \right ) \left (3x+ 7 \right )$

Correct answer:

$\left ( x+2 \right ) \left (3x- 7 \right )$

Explanation:

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is 3 (-14) = -42 and whose sum is . These numbers are -7,6 , so:

$3x^{2} - x -14$

$= 3x^{2} - 7x +6x -14$

$= \left (3x^{2} - 7x \right ) +\left ( 6x -14 \right )$

$=x \left (3x- 7 \right ) +2 \left (3x- 7 \right )$

$=\left ( x+2 \right ) \left (3x- 7 \right )$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Variables and Exponents

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #21 : Variables And Exponents

Example Question #22 : Variables And Exponents

Example Question #23 : Variables And Exponents

Example Question #24 : Variables And Exponents

Example Question #21 : Variables And Exponents

Example Question #22 : Variables And Exponents

Example Question #23 : Variables And Exponents

Example Question #1 : How To Multiply Exponential Variables

Example Question #2 : How To Multiply Exponential Variables

Example Question #3 : How To Multiply Exponential Variables

All ISEE Upper Level Quantitative Resources

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