Exponential Operations - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that .

Column A Column B

Answer

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is since . Column B is since . We can see that Column B is greater.

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Question

Which is the greater quantity?

(A) The sum of the first ten perfect square integers

(B) The sum of the first five perfect cube integers

Answer

The sum of the first ten perfect square integers:

$\begin{matrix} ;; ; 1\ ;; ; 4\ ;; ; 9\ ;16\ ;25\ ;36\ ;49\ ;64\ ;81\ \underline{100}\ 385 \end{matrix}$

The sum of the first five perfect cube integers:

$\begin{matrix} ;; ; 1\ ;; ; 8\ ;27\ ;64\ \underline{125}\ 225 \end{matrix}$

(A) is greater.

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Question

Add all of the perfect squares between 50 and 100 inclusive.

Answer

The perfect squares between 50 and 100 inclusive are

$8^{2} = 64$

$9^{2} = 81$

$10 ^{2} = 100$

Their sum is

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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

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

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: $\left (y ^{3} \right )^{4}$

Answer

Apply the power of a power property:

$\left (y ^{3} \right )^{4} = y ^{3 \cdot 4} = y^{12}$

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Question

Simplify the expression: $\left (5 a^{3}y ^{4} \right )^{2}$

Answer

Apply the power of a product rule, then apply the power of a power rule:

$\left (5 a^{3}y ^{4} \right )^{2}$

$= 5^{2} \cdot \left (a^{3}\right )^{2} \cdot \left ( y ^{4} \right )^{2}$

$= 25 \cdot a^{ 3 \cdot 2 } \cdot y ^{ 4 \cdot 2 }$

$= 25 a^{ 6} y ^{ 8 }$

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Question

Which of the following expressions is equal to $1,234^{0}$ ?

Answer

Any nonzero number raised to the power of 0 is equal to 1.

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Question

Which quantity is greater?

(a) $(-5) ^{3}$

(b) $(-5) ^{-3}$

Answer

(a) $(-5) ^{3} = (-5) \cdot (-5) \cdot (-5) = 25 \cdot (-5) = -125$

(b) $(-5) ^{-3} = \frac{1}{(-5) ^{3}} = \frac{1}{-125} =- \frac{1}{125}$

(b) is the greater quantity.

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Question

Which is the greater quantity?

(a) $2 ^{100}$

(b) $4^{50}$

Answer

$4^{50} = \left ( 2 ^{2} \right )^{50} = 2 ^{ 2; \cdot ;50 } = 2 ^{ 100 }$

The two quantities are equal.

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Question

is positive.

Which is the greater quantity?

(a) $100 ^{2x}$

(b) $1,000 ^{x}$

Answer

Use the power of a power property:

(a) $100 ^{2x} = \left (10^{2} \right ) ^{2x} = 10^{2\cdot 2x }= 10^{4x }$

(b) $1,000 ^{x} =\left ( 1 0 ^{3} \right ) ^{x} = 1 0 ^{3x}$

Since , . Subsequently,

$100 ^{2x} = 10^{4x } > 10^{3x } = 1,000^{x }$ ,

making (a) greater

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Question

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that .

Column A Column B

$x^{4}\cdot x^2$ $\frac{x^{11}}{x^{2}}$

Answer

Column A gives simplifies to give us $\ x^{6}$ , and Column B simplifies to give us $x^{9}$ . At first glance, Column B is greater, as it would be for all answers greater than 1. However, if , the two columns are equal. Furthermore, if is negative, or a fraction, Column A is greater. Thus, since we could arrive at all three answers by using different numbers, we cannot determine the answer conclusively.

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Question

Column A Column B

$\left ( \frac{5x^2}{25y^4} \right )^{0}$

Answer

Anything raised to zero is equal to 1. Therefore, Column A has to be greater because 1 is greater than 0.

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Question

Which of the following expressions is equivalent to

$\left (\sqrt{1,001}-\sqrt{999} \right ) \left ( \sqrt{1,001}+\sqrt{999} \right )$ ?

Answer

Use the difference of squares pattern as follows:

$\left (\sqrt{1,001}-\sqrt{999} \right ) \left ( \sqrt{1,001}+\sqrt{999} \right )$

$= \left (\sqrt{1,001} \right ) ^{2} - \left (\ \sqrt{999} \right ) ^{2}$

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Question

Raise $8 \times 10 ^{-3}$ to the fourth power and give the result in scientific notation.

Answer

Use the properties of exponents to raise the number to the fourth power:

$\left (8 \times 10 ^{-3} \right ) ^{4}$

$= 8 ^{4} \times \left ( 10 ^{-3} \right ) ^{4}$

$= 4,096 \times 10 ^{-3 \times 4 }$

$= 4,096 \times 10 ^{-12 }$

This is not in scientific notation, so adjust:

$4.096 \times 10^{3} \times 10 ^{-12 }$

$= 4.096 \times 10^{3 + (-12)}$

$= 4.096 \times 10^{-9}$

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Question

Which expression is equal to 65,000?

Answer

$6.510^{4}$ is equal to

Move the decimal one place to the right for each number of the exponent with a base ten.

For example, $6.5 \ast10^{^{1}}=65$ , $6.510^{^{2}}=650$ , etc.

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Question

44,000,000 can be written in scientific notation as $a \times 10 ^{N}$ for some .

Which is the greater quantity?

(A)

(B) 8

Answer

To write 44,000,000 in scientifc notation, write the implied decimal point after the final "0", then move it left until it is after the first nonzero digit (the first "4").

$44 000 000. \Rightarrow 4.4 000 000$

This requires a displacement of seven places, so

$44,000,000 = 4.4 \times 10^{7}$

, and (B) is greater.

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