How to multiply exponents - ISEE Upper Level Quantitative Reasoning

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Question

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

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

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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 $1,234^{0}$ ?

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

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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) $4^{50}$

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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) $1,000 ^{x}$

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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 $x\neq0$ .

Column A Column B

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

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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

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

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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

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

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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.

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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?

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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

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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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Question

Give the result:

$1,000,001 ^{2} - 999,999^{2}$

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Answer

For any two numbers ,

$a^{2} - b ^{2} = (a+b) (a - b)$

We can solve this problem most easily by taking advantage of this pattern, setting

:

$1,000,001 ^{2} - 999,999^{2}$

$= \left ( 1,000,001+ 999,999 \right )\left ( 1,000,001- 999,999 \right )$

$= 2,000,000 \times 2 = 4,000,000$

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Question

Give the result:

$333,335 ^{2} - 333,331 ^{2}$

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Answer

For any two numbers ,

$a^{2} - b ^{2} = (a+b) (a - b)$

We can solve this problem most easily by taking advantage of this pattern, setting

:

$333,335 ^{2} - 333,331 ^{2}$

$= \left (333,335 + 333,331 \right )\left (333,335 - 333,331 \right )$

$=666,666 \times 4$

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Question

Give the cube of $6 \times 10 ^{4}$ in scientific notation.

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Answer

$\left (6 \times 10 ^{4} \right ) ^{3}$

$= 6^{3} \times \left (10 ^{4} \right ) ^{3}$

$=216 \times 10 ^{4 \times 3 }$

$=216 \times 10 ^{12 }$

This is not in scientific notation, so adjust:

$2.16 \times 10 ^{2 } \times 10 ^{12 }$

$=2.16 \times 10 ^{2 + 12 }$

$=2.16 \times 10 ^{14 }$

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Question

$x^{3} = - 7$

$y ^{6} = 13$

Evaluate $(xy) ^{6}$ .

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Answer

By the Power of a Product Principle,

$(xy) ^{6} = x ^{6} \cdot y ^{6}$

Also, by the Power of a Power Principle,

$(x^{3} ) ^{2} = x ^{3 \cdot 2 } = x ^{6}$

Combining these ideas,

$(xy) ^{6} = (x^{3} ) ^{2} \cdot y ^{6} = (-7 ) ^{2} \cdot 13 = 7 ^{2} \cdot 13 = 49 \cdot 13 = 637$

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Question

$\dpi{100} (7-4)^{3}=$

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Answer

Subtract the numbers inside the parentheses first. This leaves you with 3, which you then raise to the 3rd power:

$\dpi{100} 3\times 3\times 3=27$

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Question

Evaluate:

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}}$

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Answer

Any nonzero number taken to the power of 0 is equal to 1, so

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}} = \frac{1+1+1}{1} = \frac{3}{1} = 3$

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Question

Multiply:

$(4r^{2} - 7) (5r^{2}+ 8 )$

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Answer

Use the FOIL method:

$(4r^{2} - 7) (5r^{2}+ 8 )$

$=4r^{2} \cdot 5r^{2} + 4r^{2} \cdot 8 -7 \cdot 5r^{2} -7 \cdot 8$

$=4\cdot 5 \cdot r^{2}\cdot r^{2} + 4\cdot 8\cdot r^{2} -7 \cdot 5\cdot r^{2} -7 \cdot 8$

$=20 \cdot r^{2+2} + 32r^{2} -35r^{2} -56$

$=20r^{4} +\left ( 32-35 \right )r^{2} -56$

$=20r^{4} -3r^{2} -56$

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Question

Simplify:

$(x^{2\sqrt{2}})^{3\sqrt{2}}$

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Answer

Based on the power rule, we know that in order to raise a power to a power we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(x^{2\sqrt{2}})^{3\sqrt{2}}=x^{2\sqrt{2}\times 3\sqrt{2}}=x^{2\times 3\times 2}=x^{12}$

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