How to add exponential variables

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ISEE Upper Level Quantitative Reasoning › How to add exponential variables

Questions 1 - 10

1

Simplify:

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

$\frac{4}{x^{2}y^{2}}$

$\frac{4}{x^{4}y^{4}}$

$\frac{4}{xy}$

$\frac{4}{x^{3}y^{3}}$

Explanation

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) - (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{(x^{4}-x^{4}+2x^{2}y^{2}+2x^{2}y^{2}+y^{4}-y^{4}) }{x^{4}y^{4}}$

$=\frac{4x^{2}y^{2}}{x^{4}y^{4}}$

$=\frac{4}{x^{2}y^{2}}$

2

Rewrite the polynomial in standard form:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

$7-2x- 3x ^{2}- 4x^{3}+ 9x^{4} + 8x^{5}$

$9x^{4} + 8x^{5} + 7- 4x^{3} - 3x ^{2} -2x$

$-2x - 3x ^{2}- 4x^{3}+ 7+ 8x^{5} + 9x^{4}$

$9x^{4} + 8x^{5} + 7-2x - 3x ^{2} - 4x^{3}$

Explanation

The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the and constant terms with exponents 1 and 0, respectively:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$= 9x^{4} - 3x ^{2} + 8x^{5} -2x ^{1} + 7x ^{0} - 4x^{3}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x ^{1}+ 7x^{0}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x+ 7$

3

Which is the greater quantity?

(a) $x^{2} + 3x$

(b)

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation

We give at least one positive value of for which (a) is greater and at least one positive value of for which (b) is greater.

Case 1:

(a) $x^{2} + 3x = 2^{2} + 3 \cdot 2 = 4 + 6 = 10$

(b) $4x = 4 \cdot 2 = 8$

Case 2: $x = \frac{1}{2}$

(a) $x^{2} + 3x = \left ( \frac{1}{2} \right ) ^{2} + 3 \cdot \frac{1}{2} = \frac{1}{4} + \frac{3}{2} = \frac{7}{4}= 1 \frac{3}{4}$

(b) $4x = 4 \cdot \frac{1}{2} = 2$

Therefore, either (a) or (b) can be greater.

4

Which is the greater quantity?

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

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

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

Explanation

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

Since and have different signs,

, and, subsequently,

Therefore,

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

This makes (b) the greater quantity.

5

Add:

Explanation

6

Define

What is $g(x ^{2} + 3x)$ ?

$2 x ^{2} + 6x - 5$

$2 x ^{2} + 6x - 10$

$2 x ^{2} + 3x - 5$

$x ^{2} + 6x - 5$

Explanation

Substitute $x ^{2} + 3x$ for in the definition:

$g(x ^{2} + 3x ) = 2 \left ( x ^{2} + 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 \left ( x ^{2}\right ) + 2 \left ( 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 x ^{2} + 6x - 5$

7

Assume that and are not both zero. Which is the greater quantity?

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

(b)

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Explanation

Simplify the expression in (a):

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

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

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

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

Therefore, whether (a) or (b) is greater depends on the values of and , neither of which are known.

8

Assume all variables to be nonzero.

Simplify: $\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0}$

None of the answer choices are correct.

$15x^{5}y ^{4}z^{3}$

$36x^{5}y ^{4}z^{3}$

$15x^{10}y ^{8}z^{6}$

$36x^{10}y ^{8}z^{6}$

Explanation

Any nonzero expression raised to the power of 0 is equal to 1. Therefore,

$\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0} = 1 + 1 = 2$ .

None of the given expressions are correct.

9

Simplify:

Explanation

Expand each term by using FOIL:

Rearrange to group like-terms together.

Simplify by combining like-terms.

10

Assume that . Simplify:

$\frac{(x^{2}+y^{2})^{2} +(x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

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

$\frac{2x^{4}-2y^{4} }{x^{4}y^{4}}$

$\frac{4}{x^{2}y^{2}}$

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

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

Explanation

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

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) + (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{x^{4}+x^{4}+2x^{2}y^{2}-2x^{2}y^{2}+y^{4}+y^{4} }{x^{4}y^{4}}$

$=\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

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