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ISEE Upper Level Math : Variables

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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 #294 : Algebraic Concepts

Simplify:

$(0.5x-0.4)^{2}$

Possible Answers:

$2.5x^{2} + 0.4x- 1.6$

$0.25x^{2}+ 0.4x- 0.16$

$2.5x^{2} - 0.4x+ 1.6$

$0.25x^{2} - 0.4x+ 0.16$

$0.25x^{2} - 0.16$

Correct answer:

$0.25x^{2} - 0.4x+ 0.16$

Explanation:

Use the pattern, substituting A = 0.5x,B = 0.4 .

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

$(0.5x - 0.4)^{2} = 0.5x^{2} - 2\cdot 0.5x \cdot 0.4+ 0.4^{2}$

$= 0.25x^{2} - 0.4x+ 0.16$

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Example Question #15 : How To Multiply Exponential Variables

Write in expanded form.

$(x\neq 0)$

$(4x+\frac{3}{x})^3$

Possible Answers:

$64x^3+144x+\frac{108}{x}-\frac{27}{x^3}$

$64x^3-144x+\frac{108}{x}-\frac{27}{x^3}$

$64x^3+144x+\frac{108}{x}+\frac{27}{x^2}$

$64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

$64x^3+172x+\frac{108}{x}+\frac{27}{x^3}$

Correct answer:

$64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

Explanation:

$(4x+\frac{3}{x})^3$

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

$=4^3\times x^3+3\times 4^2\times 3\times \frac{x^2}{x}+3\times 4\times 3^2\times \frac{x}{x^2}+\frac{3^3}{x^3}$

$=64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

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Example Question #301 : Algebraic Concepts

Simplify:

2(x^2y^3)^2(3xy^2)^3

Possible Answers:

$27x^7y^{13}$

$x^7y^{12}$

$2x^7y^{13}$

$54x^7y^{12}$

Correct answer:

$54x^7y^{12}$

Explanation:

First, simplify all of the exponents. When the exponent is outside of the parantheses, multiply it by the exponents inside so that you get: 2(x^4y^6)(3^3x^3y^6) . Multiply 3^3 so that you get 27. Then, multiply like terms. First, multilpy 2 by 27 so that you get 54. Then, multiply the x terms. Remember, when bases are the same, add the exponents: $x^4\cdot x^3=x^7$ . Then, multiply the y terms: $y^6\cdot y^6=y^{12}$ . Then, multiply all of the terms together so that you get $54x^7y^{12}$ .

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Example Question #302 : Algebraic Concepts

Simplify:

$\frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot xy$

Possible Answers:

$\frac{ x^{3 } y^{4} }{2 }$

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

$\frac{ x^{3 } y^{5} }{2 }$

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

Correct answer:

$\frac{ x^{3 } y^{5} }{2 }$

Explanation:

$\frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot xy$

$= \frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot \frac{xy}{1}$

$= \frac{4x^{3} \cdot y^{5} \cdot xy }{y \cdot 8x \cdot 1 }$

$= \frac{4x^{3+1 } \cdot y^{5+1} }{8xy }$

$= \frac{4x^{4 } \cdot y^{6} }{8xy }$

$= \frac{ x^{4-1 } \cdot y^{6-1} }{2 }$

$= \frac{ x^{3 } y^{5} }{2 }$

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Example Question #303 : Algebraic Concepts

Simplify the following expression:

(9x^5y^2)(12x^3y^8)

Possible Answers:

$108x^8y^{16}$

$108x^8y^{10}$

$108x^{15}y^{10}$

$98x^8y^{10}$

Correct answer:

$108x^8y^{10}$

Explanation:

Simplify the following expression:

(9x^5y^2)(12x^3y^8)

To combine these, we need to multiply our coefficients and our variables.

First, multiply the coefficients

9*12=108

Next, multiply our variables by adding the exponent:

$(x^5y^2)(x^3y^8)=x^{5+3}y^{2+8}=x^8y^{10}$

So, we put it all together to get:

$108x^8y^{10}$

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Example Question #111 : Variables

Simplify the following:

$a^2 \cdot a^4$

Possible Answers:

a^4

a^8

a^2

$a^{12}$

a^6

Correct answer:

a^6

Explanation:

When multiply variables with exponents, we will use the following formula:

$a^x \cdot a^y = a^{x+y}$

So, we can write the problem like this:

$a^2 \cdot a^4 = a^{2+4}$

$a^2 \cdot a^4 = a^6$

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Example Question #301 : Algebraic Concepts

Combine the following:

$x^3 \cdot x^7$

Possible Answers:

$x^{10}$

$x^{21}$

$x^{37}$

x^4

$2x^{10}$

Correct answer:

$x^{10}$

Explanation:

To combine variables with exponents, we will use the following formula:

$x^a \cdot x^b = x^{a+b}$

Now, let’s combine the following:

$x^3 \cdot x^7 = x^{3+7}$

$x^3 \cdot x^7 = x^{10}$

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Example Question #303 : Algebraic Concepts

Combine the following:

$a^5 \cdot b^2 \cdot a^7 \cdot b^4$

Possible Answers:

$a^{12} \cdot b^6$

$ab^{18}$

$ab^{43}$

$2a^{12} \cdot 2b^6$

$a^{35} \cdot b^8$

Correct answer:

$a^{12} \cdot b^6$

Explanation:

When combining variables with exponents, we can only combine like variables.

When multiply variables with exponents, we add the exponents. In other words:

$a^x \cdot a^y = a^{x+y}$

So, we get

$a^5 \cdot b^2 \cdot a^7 \cdot b^4$

$a^5 \cdot a^7 \cdot b^2 \cdot b^4$

$a^{5+7} \cdot b^{2+4}$

$a^{12} \cdot b^6$

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Example Question #112 : Variables

Simplify:

$b^2 \cdot b^7$

Possible Answers:

b^5

$b^{27}$

$b^{14}$

b^9

b^7

Correct answer:

b^9

Explanation:

To simplify, we will use the following formula:

$x^a \cdot x^b = x^{a+b}$

So, given the problem, we get

$b^2 \cdot b^7 = b^{2+7}$

$b^2 \cdot b^7 = b^9$

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Example Question #55 : Variables And Exponents

Combine the following:

$b^4 \cdot b^5 \cdot b$

Possible Answers:

$b^{10}$

$3b^{10}$

b^9

$b^{20}$

3b^9

Correct answer:

$b^{10}$

Explanation:

To multiply terms with exponents, we will use the following formula:

$a^x \cdot a^y = a^{x+y}$

So, we get

$b^4 \cdot b^5 \cdot b$

$b^4 \cdot b^5 \cdot b^1$

$b^{4+5+1}$

$b^{10}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Variables

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #294 : Algebraic Concepts

Example Question #15 : How To Multiply Exponential Variables

Example Question #301 : Algebraic Concepts

Example Question #302 : Algebraic Concepts

Example Question #303 : Algebraic Concepts

Example Question #111 : Variables

Example Question #301 : Algebraic Concepts

Example Question #303 : Algebraic Concepts

Example Question #112 : Variables

Example Question #55 : Variables And Exponents

All ISEE Upper Level Math Resources

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