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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

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Example Question #1 :How To Divide Exponents

Simplify:

$\frac{4x^8}{2x^0}$

Possible Answers:

2x^7

x^7

x^8

2x^9

2x^8

Correct answer:

2x^8

Explanation:

Based on the quotient rule for exponents, we know that for any non-zero numberand any integersand,

$\frac{x^a}{x^b}=x^{a-b}$ .

我们也知道,任何非零的数字raised to the power ofis.

$\frac{4x^8}{2x^0}=\frac{4}{2}\times \frac{x^8}{1}=2x^8$

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Example Question #2 :How To Divide Exponents

Simplify:

$\frac{6x^4-6}{3x^2+3x^0}$

Possible Answers:

x^2-1

2(x+1)

2(x-1)

2(x^2+1)

2(x^2-1)

Correct answer:

2(x^2-1)

Explanation:

We know that any non-zero number raised to the power ofis equal to.

$\frac{6x^4-6}{3x^2+3x^0}=\frac{6(x^4-1)}{3(x^2+x^0)}=\frac{6(x^4-1)}{3(x^2+1)}$

Now factor:

$\frac{6(x^4-1)}{3(x^2+1)}=\frac{6}{3}\frac{(x^2-1)(x^2+1)}{x^2+1}=2(x^2-1)$

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Example Question #31 :Exponents

Evaluate:

$\frac{4^0}{2\times 5^0-12^0}$

Possible Answers:

$-\frac{2}{5}$

The expression is undefined.

Correct answer:

Explanation:

We know that any non-zero number raised to the power ofis equal to.

$\frac{4^0}{2\times 5^0-12^0}=\frac{1}{2\times 1-1}=\frac{1}{2-1}=1$

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Example Question #92 :Numbers And Operations

Evaluate:

$\frac{y^{-2}}{y^{-5}}$

Possible Answers:

$y^{-3}$

$y^{-2}$

y^3

y^2

Correct answer:

y^3

Explanation:

Based on the quotient rule for exponents, we know that, for any non-zero numberand any integersand,

$\frac{x^a}{x^b}=x^{a-b}$ .

$\frac{y^{-2}}{y^{-5}}=y^{-2-(-5)}=y^{-2+5}=y^{3}$

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例子问题# 93:Numbers And Operations

Simplify:

$(\frac{3x}{y^3})^{3}$

Possible Answers:

$\frac{27x^3}{y^3}$

$\frac{27x^3}{y^9}$

$\frac{9x^3}{y^9}$

$\frac{18x^3}{y^9}$

$\frac{27x^3}{y^6}$

Correct answer:

$\frac{27x^3}{y^9}$

Explanation:

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

$(\frac{3x}{y^3})^{3}=\frac{(3x)^3}{(y^3)^3}$

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ ,

whereandare real numbers.

$(\frac{3x}{y^3})^{3}=\frac{(3x)^3}{(y^3)^3}=\frac{(3^{3}\times x^3)}{y^{3\times 3}}=\frac{27x^3}{y^9}$

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Example Question #94 :Numbers And Operations

Evaluate:

$(\frac{x}{y^4})^{5}$

Possible Answers:

$\frac{x^5}{y^{10}}$

$\frac{x^5}{y^{20}}$

$\frac{x^5}{y^{9}}$

$\frac{x}{y^{20}}$

$\frac{x^4}{y^{20}}$

Correct answer:

$\frac{x^5}{y^{20}}$

Explanation:

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

$(\frac{x}{y^4})^{5}=\frac{x^5}{(y^4)^5}$

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(\frac{x}{y^4})^{5}=\frac{x^5}{(y^4)^5}=\frac{x^5}{y^{4\times 5}}=\frac{x^5}{y^{20}}$

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Example Question #531 :Isee Upper Level (Grades 9 12) Mathematics Achievement

Simplify:

$\frac{x^{-4}}{y^{-8}}$

Possible Answers:

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

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

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

$\frac{y^8}{x}$

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

Correct answer:

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

Explanation:

The Negative Exponent Rule says $a^{-n}=\frac{1}{a^n}$ .

$\frac{x^{-4}}{y^{-8}}=\frac{y^8}{x^4}$

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Example Question #532 :Isee Upper Level (Grades 9 12) Mathematics Achievement

Evaluate:

$(\frac{14x^2}{7x})^0$

Assume variables are non-zero.

Possible Answers:

2x^2

Correct answer:

Explanation:

We know that any non-zero number raised to the power ofis equal to.

$(\frac{14x^2}{7x})^0=1$

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Example Question #97 :Numbers And Operations

Evaluate:

$(\frac{22x^3}{11y})^5$

Possible Answers:

$\frac{32x^{5}}{y^5}$

$\frac{32x^{15}}{y^5}$

$\frac{16x^{15}}{y^5}$

$\frac{2x^{15}}{y^5}$

$\frac{32x^{10}}{y^5}$

Correct answer:

$\frac{32x^{15}}{y^5}$

Explanation:

When a fraction is raised to a power, the numerator and the denominator are both raised to that power. In addition, the power rule says that in order to raise a power to a power, multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(\frac{22x^3}{11y})^{5}=\frac{(22x^3)^{5}}{(11y)^5}=(\frac{22}{11})^5\times \frac{x^{3\times 5}}{y^5}=2^5\times \frac{x^{15}}{y^5}=\frac{32x^{15}}{y^5}$

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Example Question #98 :Numbers And Operations

What is the value of the expression $\frac{(4.2\times 10^6)}{(7.0\times 10^{-2})}$ ?

Possible Answers:

$6.0\times 10^6$

$6.0\times 10^8$

$5.0\times 10^7$

$6.0\times 10^7$

$6.0\times 10^{-6}$

Correct answer:

$6.0\times 10^7$

Explanation:

When dealing with dividing scientific notation, you can divide the like terms separately. First, divide 4.2 by 7.0. This yields 0.6. Then divide the 10s. $10^6\div 10^{-2}=10^8$ because when you divide a term with the same base, you subtract exponents. This gives you a combined answer of $0.6\times 10^8$ . Since this is not in scientific notation, move the decimal over one place to the right so that it will be in the appropriate notation. This gives you $6.0\times 10^7$ .

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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 #1 :How To Divide Exponents

Example Question #2 :How To Divide Exponents

Example Question #31 :Exponents

Example Question #92 :Numbers And Operations

例子问题# 93:Numbers And Operations

Example Question #94 :Numbers And Operations

Example Question #531 :Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #532 :Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #97 :Numbers And Operations

Example Question #98 :Numbers And Operations

All ISEE Upper Level Math Resources

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