How to divide exponents - ISEE Upper Level Quantitative Reasoning

Card 0 of 112

Question

Column A Column B

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

Answer

You can simplify Column A first. When you're dividing with exponents and bases are the same, subtract the exponents. Therefore, it simplifies to x. We know that x is positive since it is greater than 1. X is greater than $\sqrt{x}$ . Try plugging in a number to test. 25 is greater than $\sqrt{25}$ , which is 5. Even 1.1 is greater than $\sqrt{1.1}$ . Therefore, Column A is greater.

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Question

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}}$

Answer

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}} = \frac{ \left ( 30 -30\right )^{0}}{15^{0}}=\frac{ 0^{0}}{15^{0}}$

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

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Question

Column A Column B

$\left(\frac{1}{16}\right)^\frac{1}{2}$ $\left(\frac{1}{16}\right)^{-\frac{1}{2}}$

Answer

Let's simplify both quantities first before we compare them. $\left(\frac{1}{16}\right)^\frac{1}{2}$ becomes $\sqrt{\frac{1}{16}}$ because the fractional exponent indicates a square root. We can simplify that by knowing that we can take the square roots of both the numerator and denominator, as shown by: $\frac{\sqrt{1}}{\sqrt{16}}$ . We can simplify further by taking the square roots (they're perfect squares) and get $\frac{1}{4}$ . Then, let's simplify Column B. To get rid of the negative exponent, we put the numerical expression on the denominator. There's still the fractional exponent at play, so we'll have a square root as well. It looks like this now: $\frac{1}{\sqrt{\frac{1}{16}}}$ . We already simplified $\sqrt{\frac{1}{16}}$ , so we can just plug in our answer, $\frac{1}{4}$ , into the denominator. Since we don't want a fraction in the denominator, we can multiply by the reciprocal of $\frac{1}{4}$ , which is 4 to get , which is just 4. Therefore, Column B is greater.

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Question

Give the reciprocal of $1.6 \times 10 ^{-7}$ in scientific notation.

Answer

The reciprocal of $1.6 \times 10 ^{-7}$ is the quotient of 1 and the number;

$\frac{1}{ 1.6 \times 10 ^{-7} }$

$= \frac{1 \times 10^{0}}{ 1.6 \times 10 ^{-7} }$

$= \frac{1 }{ 1.6 } \times \frac{ 10^{0}}{ 10 ^{-7} }$

$=0.625 \times 10^{0- (-7) }$

$=0.625 \times 10^{7 }$

This is not in scientific notation, so adjust.

$6.25 \times 10^{-1 } \times 10^{7 }$

$= 6.25 \times 10^{-1+7 }$

$= 6.25 \times 10^{6 }$

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Question

Give the reciprocal of $3.2 \times 10 ^{9}$ in scientific notation.

Answer

The reciprocal of $3.2 \times 10 ^{9}$ is the quotient of 1 and the number, or

$\frac{1}{ 3.2 \times 10 ^{9} }$

$= \frac{1 \times 10^{0}}{ 3.2 \times 10 ^{9}}$

$= \frac{1 }{ 3.2 } \times \frac{ 10^{0}}{ 10 ^{9} }$

$=0.3125 \times 10 ^{-9}$

This is not in scientific notation, so adjust:

$3.125 \times 10 ^{-1}\times 10 ^{-9}$

$=3.125 \times 10 ^{-1+ (-9)}$

$=3.125 \times 10 ^{-10}$

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Question

Simplify: $\frac{x^{50}}{x^{10}}$

Answer

Apply the quotient of powers rule.

$\frac{x^{50}}{x^{10}} = x^{50-10} = x^{40}$

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Question

Simplify: $\frac{12x^{12}}{3x^{3}}$

Answer

Separate the fraction and apply the quotient of powers rule:

$\frac{12x^{12}}{3x^{3}} = \frac{12}{3} \cdot \frac{x^{12}}{x^{3}} = 4x^{12-3} = 4x^{9}$

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Question

Evaluate:

$\frac{120}{12^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $12^{0} = 1$ , and

$\frac{120}{12^{0}} = \frac{120}{1} = 120$

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Question

Evaluate:

$\frac{25}{10 ^{0} -5^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $5^{0} = 10 ^{0} =1$ . Therefore,

$\frac{25}{10 ^{0} -5^{0}} = \frac{25}{1-1} = \frac{25}{0}$

However, an expression with a denominator of 0 is undefined, so that is the correct choice.

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Question

Simplify:

$\frac{6x^6}{2x^3}$

Answer

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

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

We should separate the fraction and apply the quotient rule for exponents:

$\frac{6x^6}{2x^3}=\frac{6}{2}\times \frac{x^6}{x^3}=3\times x^{6-3}$

$\Rightarrow \frac{6x^6}{2x^3}=3x^3$

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Question

Simplify:

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

Answer

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

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

We also know that any non-zero number raised to the power of is .

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

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Question

Simplify:

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

Answer

We know that any non-zero number raised to the power of is 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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Question

Evaluate:

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

Answer

We know that any non-zero number raised to the power of is 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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Question

Evaluate:

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

Answer

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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Question

Simplify:

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

Answer

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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Question

Simplify the following:

$\frac{18t^6u^{14}}{6t^3u^{12}}$

Answer

Simplify the following:

$\frac{18t^6u^{14}}{6t^3u^{12}}$

Let's begin by recalling that when dividing variables with similar base, we need to subtract the exponents.

To deal with the coefficients, simply treat them as a fraction and simplify.

$\frac{18t^6u^{14}}{6t^3u^{12}}=\frac{18}{6}t^{6-3}u^{14-12}=3t^3u^2$

So our answer is

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Question

Evaluate:

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

Answer

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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Question

Simplify:

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

Answer

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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Question

Evaluate:

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

Assume variables are non-zero.

Answer

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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Question

Evaluate:

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

Answer

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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