How to divide exponential variables - ISEE Upper Level Quantitative Reasoning

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Question

Half of one hundred divided by five and multiplied by one-tenth is __________.

Answer

Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.

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Question

Simplify: $\frac{8x^{-6}}{2x^{-3}}$

Answer

$\frac{8x^{-6}}{2x^{-3}} = \frac{8}{2} \cdot \frac{x^{-6}}{x^{-3}} = 4 \cdot x ^{-6 - (-3)} = 4 \cdot x ^{-3} =4 \cdot \frac{1}{x^{3}} = \frac{4}{x^{3}}$

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Question

Simplify:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

Answer

Break the fraction up and apply the quotient of powers rule:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

$= \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$

$= \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$

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

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

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

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Question

Simplify: $\frac{16x^4}{12x^2}$

Answer

To simplify this expression, look at the like terms separately. First, simplify $\frac{16}{12}$ . This becomes $\frac{4}{3}$ . Then, deal with the $\frac{x^4}{x^2}$ . Since the bases are the same and you're dividing, you can subtract exponents. This gives you Since the exponent is positive, you put in the numerator. This gives you a final answer of $\frac{4x^2}{3}$ .

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Question

is a negative number.

Which is the greater quantity?

(a) The reciprocal of $x^{7}$

(b) The reciprocal of $x^{8}$

Answer

A negative number raised to an odd power is negative; a negative number raised to an even power is positive. It follows that $x^{7}$ is negative and $x^{8}$ is positive. Also, the reciprocal of a nonzero number assumes the same sign as the number itself, so the reciprocal of $x^{8}$ is positive and that of $x^{7}$ is negative. It follows that the reciprocal of $x^{8}$ is the greater of the two.

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Question

Simplify:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

Answer

Break the fraction up and apply the quotient of powers rule:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

$= \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$

$= \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$

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

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

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

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Question

Simplify the following:

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

Answer

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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Question

Simplify the following:

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

Answer

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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Question

If , simplify:

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

Answer

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

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

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

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

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Question

Simplify:

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

Answer

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

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Question

Simplify if $xy eq \pm 2$ .

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

Answer

Start by factoring the numerator. 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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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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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Question

Simplify:

$\frac{(2xy)^{6}}{2x^{2} y^{8}}$

Answer

$\frac{(2xy)^{6}}{2x^{2} y^{8}}$

$= \frac{2^{6} \cdot x^{6}y^{6}}{2x^{2} y^{8}}$

$= \frac{64 x^{6-2}}{2 y^{8-6}}$

$= \frac{32 x^{4} }{ y^{2}}$

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Question

Divide the following:

$x^8 \div x^2$

Answer

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

$x^a \div x^b = x^{a-b}$

Now, let’s combine the following:

$x^8 \cdot x^2 = x^{8-2}$

$x^8 \cdot x^2 = x^{6}$

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Question

Divide the following:

$\frac{a^2 b^4}{a b^2}$

Answer

When dividing variables, we can only divide like variable.

Also, when dividing variables with exponents, we subtract the exponents. In other words:

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{a^2 b^4}{a b^2}$

$\frac{a^2b^4}{a^1b^2}$

$a^{2-1}b^{4-2}$

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Question

Simplify:

$y^{12} \div y^2$

Answer

To simplify, we will use the following formula:

$x^a \div x^b = x^{a-b}$

So, given the problem, we get

$y^{12} \div y^2 = y^{12-2}$

$y^{12} \div y^2 = y^{10}$

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Question

Divide the following:

$\frac{h^{10}}{h^2}$

Answer

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

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{h^{10}}{h^2}$

$h^{10-2}$

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Question

Divide the following:

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

Answer

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

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{y^9}{y^3} = y^{9-3}$

$\frac{y^9}{y^3} = y^6$

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Question

Solve the following:

$\frac{18x^4}{9x}$

Answer

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

$\frac{a^x}{a^y} = a^{x-y}$

Also, we divide the coefficients like normal.

So, we get

$\frac{18x^4}{9x}$

$\frac{2x^4}{1x}$

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

$\frac{2x^4}{x^1}$

$2x^{4-1}$

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