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ISEE Upper Level Math : Numbers and Operations

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

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #16 : How To Divide Exponents

What is a simplified version of the expression below?

$\frac{8x^{9}}{4x}$

Possible Answers:

$2^{8}$

$2x^{4}$

$2x^{8}$

$x^{18}$

Correct answer:

$2x^{8}$

Explanation:

When exponents are divided, the exponent in the denominator is subtracted from the exponent in the numerator. Given that the exponent in the numerator is , and the exponent in the denominator is , the resulting exponent will be because 9-1=8 . Thus, will be raised to an exponent of .

Meanwhile, will be divided by , resulting in .

Therefore, the answer is $2x^{8}$ .

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Example Question #101 : Numbers And Operations

What is the value of the expression below?

$\frac{x^{6}}{x^{3}}$

Possible Answers:

$x^{3}$

$x^{2}$

$\frac{3x}{x}$

Correct answer:

$x^{3}$

Explanation:

When values having the same base are divided, their exponents are subtracted.

$\frac{x^{6}}{x^{3}}$

6-3=3

Therefore, the correct answer is

$x^{3}$

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Example Question #102 : Numbers And Operations

Simplify:

$\frac{\left ( 4 st^{2} \right )^{4}}{s^{4} \cdot 8 t}$

Possible Answers:

$32 t^{8}$

$32s t^{8}$

$32 s t^{7}$

$32 t^{7}$

Correct answer:

$32 t^{7}$

Explanation:

$\frac{\left ( 4 st^{2} \right )^{4}}{s^{4} \cdot 8 t}$

$= \frac{4^{4} \cdot s^{4} \left ( t^{2} \right )^{4}}{8 s^{4} t^{1}}$

$= \frac{256 s^{4} \cdot t^{2 \cdot 4 } }{8 s^{4} t^{1}}$

$= \frac{256 s^{4} t^{8 } }{8 s^{4} t^{1}}$

$= \frac{32 t^{8 } }{ t^{1}}$

$=32 t^{8-1}$

$=32 t^{7}$

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Example Question #21 : How To Divide Exponents

Simplify the following expression:

$\frac{16r^{19}}{8r^6}$

Possible Answers:

2r^5

$2r^{13}$

$8r^{13}$

$2r^{\frac{19}{6}}$

Correct answer:

$2r^{13}$

Explanation:

Simplify the following expression:

$\frac{16r^{19}}{8r^6}$

To solve this question, we need to recall that when dividing exponents we subtract them.

When dividing coefficients, we treat them as regular division.

In this case, we can break up our question into two parts:

$\frac{16r^{19}}{8r^6}\rightarrow\frac{16}{8} \frac{r^{19}}{r^6}$

The coefficients will simply reduce to "2", because 16 divided by 8 is two

The r's are another story. We subtract, so we will get the following:

$\frac{r^{19}}{r^6}=r^{19-6}=r^{13}$

So, if we put our two parts back together, we get:

$2r^{13}$

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Example Question #51 : Exponents

Simplify the following:

$\frac{18t^6u^{14}}{6t^3u^{12}}$

Possible Answers:

3t^2u^2

12t^3u^2

3t^3u^3

3t^3u^2

Correct answer:

3t^3u^2

Explanation:

Simplify the following:

$\frac{18t^6u^{14}}{6t^3u^{12}}$

Let's begin by recalling that when dividing variables with similar base, we need to subtract the exponents.

To deal with the coefficients, simply treat them as a fraction and simplify.

$\frac{18t^6u^{14}}{6t^3u^{12}}=\frac{18}{6}t^{6-3}u^{14-12}=3t^3u^2$

So our answer is

3t^3u^2

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Example Question #102 : Numbers And Operations

Simplify the expression:

$6x^{4} + 8x^{3} - 4x^{2}$

Possible Answers:

$6x^{4} + 4x$

$2x^{2}(3x^{2}+4x-2)$

$10x^{5}$

$10x^{9}$

$6x^{4} + 4x^{5}$

Correct answer:

$2x^{2}(3x^{2}+4x-2)$

Explanation:

$6x^{4} + 8x^{3} - 4x^{2}$

$=2x^{2}(3x^{2}+4x-2)$

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Example Question #2 : How To Add Exponents

Simplify:

4x^2+8x^3+16x^2+4x^3

Possible Answers:

4x^2(5+3x^2)

4x^2(5+3x)

4x(5+3x)

4x^2(5+x)

x^2(5+3x)

Correct answer:

4x^2(5+3x)

Explanation:

In order to add exponential terms, both the base and the exponent must be the same. So we can write:

4x^2+8x^3+16x^2+4x^3=(4x^2+16x^2)+(8x^3+4x^3)

=20x^2+12x^3=4x^2(5+3x)

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Example Question #53 : Exponents

Evaluate:

$\frac{x^0+y^0+1}{x^0-y^0-1}$

Possible Answers:

Correct answer:

Explanation:

Based on the zero-exponent rule we have:

a^0=1

That means any non-zero number raised to the zero power is equal to . So we can write:

$\frac{1+1+1}{1-1-1}=\frac{3}{-1}=-3$

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Example Question #3 : How To Add Exponents

What is the value of the expression:

$x^{3}\cdotx^{y}\cdot x^{2+y}$

Possible Answers:

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

$x^{3}\cdotx^{y+2}$

$x^{4}\cdotx^{y+4}$

$x^{6}\cdotx^{y+2}$

Correct answer:

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

Explanation:

When values, having the same base, are multiplied by one another, the exponents are added together and the base stays the same.

Thus,

$x^{3}\cdotx^{y}\cdot x^{2+y}$

is equal to

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

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Example Question #4 : How To Add Exponents

Simplify: $x^{7}\cdot2x^{3}$

Possible Answers:

$x^{21}$

$2x^{10}$

$x^{10}$

$2x^{21}$

Correct answer:

$2x^{10}$

Explanation:

When multiplying exponents, the exponents are added together. Thus, 3 and 7 are added together for a sum of 10. In this problem, the "2" becomes a coefficient in front of the x. Therefore, the correct answer is:

$2x^{10}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Numbers and Operations

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #16 : How To Divide Exponents

Example Question #101 : Numbers And Operations

Example Question #102 : Numbers And Operations

Example Question #21 : How To Divide Exponents

Example Question #51 : Exponents

Example Question #102 : Numbers And Operations

Example Question #2 : How To Add Exponents

Example Question #53 : Exponents

Example Question #3 : How To Add Exponents

Example Question #4 : How To Add Exponents

All ISEE Upper Level Math Resources

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