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ISEE Upper Level Math : Variables and Exponents

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Example Questions

Example Question #23 : How To Multiply Exponential Variables

Multiply the following:

$b^4 \cdot b^2$

Possible Answers:

b^6

$b^{42}$

b^8

b^2

b^4

Correct answer:

b^6

Explanation:

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

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

Now, let’s combine the following:

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

$b^4 \cdot b^2 = b^{6}$

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Example Question #24 : How To Multiply Exponential Variables

Solve the following:

$3y^3 \cdot 4y^2$

Possible Answers:

7y^6

12y

7y^5

12y^6

12y^5

Correct answer:

12y^5

Explanation:

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

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

Also, we will multiply coefficients like normal.

So, we get

$3y^3 \cdot 4y^2$

$12(y^3 \cdot y^2)$

$12(y^{3+2})$

12y^5

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Example Question #25 : How To Multiply Exponential Variables

Simplify the following expression:

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

Possible Answers:

$98x^8y^{10}$

$108x^8y^{16}$

$108x^{10}y^{24}$

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

$108x^8y^{10}$

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 #1 : How To Divide Exponential Variables

Simplify the following:

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

Possible Answers:

$p^{9}$

$p^{2}$

$p^{3}$

$\frac{p}{2}$

Correct answer:

$p^{3}$

Explanation:

To divide a variable with an exponent, you can write out all the multiplies, which for this question would look like:

$\frac{pppppp}{ppp}$

Then cancel out the matching p's, leaving:

$ppp=p^{3}$

Or, if the top and bottom are the same variable, you can simply subtract the bottom exponent from the top exponent.

$p^{6-3}=p^{3}$

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Example Question #2 : How To Divide Exponential Variables

Simplify the following:

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

Possible Answers:

$y^{3}$

$y^{0.4}$

$\frac{1}{3y}$

$\frac{1}{y^{3}}$

Correct answer:

$\frac{1}{y^{3}}$

Explanation:

To divide variables with exponents, either:

Write out the multiplies, then reduce

$\frac{yy}{yyyyy}=\frac{1}{y^{3}}$

or subtract the bottom exponent from the top, like so

$y^{2-5}=y^{-3}=\frac{1}{y^{3}}$

Remember, negative exponent means dividing. $y^{-3}$ would also be correct, but you should know that is equivalent to $\frac{1}{y^{3}}$

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Example Question #3 : How To Divide Exponential Variables

If $x\neq -1,0,1$ , simplify:

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

Possible Answers:

3(x+1)

2(x^2+1)

3(x^2-1)

3(x-1)

3(x^2+1)

Correct answer:

3(x^2+1)

Explanation:

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

$=\frac{3x(x^2+1)\left [ (x+1)(x-1) \right ]}{x(x^2-1)}$

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

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

=3(x^2+1)

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Example Question #2 : How To Divide Exponential Variables

Simplify:

$\frac{x^6y^2z^3}{x^3y^4z^3}$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

$\frac{x^6y^2z^3}{x^3y^4z^3}=x^{6-3}y^{2-4}z^{3-3}=x^3y^{-2}z^0=\frac{x^3}{y^2}$

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Example Question #5 : How To Divide Exponential Variables

Simplify if $xy\neq \pm 2$ .

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

Possible Answers:

x^2y^2

-xy

$x^{-1}y^2$

-x^2y^2

Correct answer:

x^2y^2

Explanation:

Start by factoring the numerator. x^2y^2 can be removed from each term.

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

Next, expand the denominator.

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

Simplify by canceling terms.

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

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Example Question #6 : How To Divide Exponential Variables

Simplify:

$\frac{2x^{-7}}{(3x)^{-2}}$

Possible Answers:

$\frac{18}{x^4}$

9x^5

$\frac{18}{x^5}$

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

18x^5

Correct answer:

$\frac{18}{x^5}$

Explanation:

In the denominator, apply the exponent to the terms in parentheses.

$\frac{2x^{-7}}{(3x)^{-2}}=\frac{2x^{-7}}{(3^{-2}) (x^{-2})}$

Terms with negative exponents can be inverted within the fraction to eliminate the negative.

$\frac{2x^{-7}}{(3^{-2}) (x^{-2})}=\frac{2(3^2)(x^2)}{x^7}$

Simplify by canceling variables and mulitplying.

$\frac{2(3^2)(x^2)}{x^7}=\frac{18x^2}{x^7}=\frac{18}{x^5}$

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Example Question #7 : How To Divide Exponential Variables

Simplify:

$\frac{8x^{-2a}y^{2a}}{4x^ay^{-2a}}$

Possible Answers:

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

$2y^{4a}}{x^{3a}$

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

$y^{4a}}{x^{3a}$

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

Correct answer:

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

Explanation:

Break up the fraction into components with like-terms.

$\frac{8x^{-2a}y^{2a}}{4x^ay^{-2a}}=\frac{8}{4}\times \frac{x^{-2a}}{x^a}\times \frac{y^{2a}}{y^{-2a}}$

Simplify by combining exponents. Remember that exponents in the denominator will become negative.

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

$2x^{-3a}y^{4a}$

Finally, put any negative exponent terms into the denominator.

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

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Variables and Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #23 : How To Multiply Exponential Variables

Example Question #24 : How To Multiply Exponential Variables

Example Question #25 : How To Multiply Exponential Variables

Example Question #1 : How To Divide Exponential Variables

Example Question #2 : How To Divide Exponential Variables

Example Question #3 : How To Divide Exponential Variables

Example Question #2 : How To Divide Exponential Variables

Example Question #5 : How To Divide Exponential Variables

Example Question #6 : How To Divide Exponential Variables

Example Question #7 : How To Divide Exponential Variables

All ISEE Upper Level Math Resources

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