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ISEE Upper Level Math : Exponents

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 #41 : Exponents

Evaluate:

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

Possible Answers:

$y^{-2}$

y^3

y^2

$y^{-3}$

Correct answer:

y^3

Explanation:

Based on the quotient rule for exponents, we know that, for any non-zero number and any integers and ,

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

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

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Example Question #42 : Exponents

Simplify:

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

Possible Answers:

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

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

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

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

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

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.

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

where and are 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 #43 : Exponents

Evaluate:

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

Possible Answers:

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

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

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

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

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

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 #41 : Exponential Operations

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #42 : Exponential Operations

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 of is equal to .

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

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Example Question #46 : Exponents

Evaluate:

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

Possible Answers:

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

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

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

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

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

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 #47 : Exponents

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

Possible Answers:

$6.0\times 10^{-6}$

$5.0\times 10^7$

$6.0\times 10^8$

$6.0\times 10^6$

$6.0\times 10^7$

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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Example Question #48 : Exponents

What is a simplified version of the expression below?

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

Possible Answers:

$x^{18}$

$2x^{8}$

$2x^{4}$

$2^{8}$

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 #43 : Exponential Operations

What is the value of the expression below?

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

Possible Answers:

$x^{2}$

$x^{3}$

$\frac{3x}{x}$

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 #50 : Exponents

Simplify:

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

Possible Answers:

$32 t^{7}$

$32s t^{8}$

$32 t^{8}$

$32 s t^{7}$

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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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #41 : Exponents

Example Question #42 : Exponents

Example Question #43 : Exponents

Example Question #41 : Exponential Operations

Example Question #42 : Exponential Operations

Example Question #46 : Exponents

Example Question #47 : Exponents

Example Question #48 : Exponents

Example Question #43 : Exponential Operations

Example Question #50 : Exponents

All ISEE Upper Level Math Resources

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