How to multiply exponents - ISEE Upper Level Quantitative Reasoning
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Simplify: 
Simplify:
Apply the power of a power property:

Apply the power of a power property:
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Simplify the expression: 
Simplify the expression:
Apply the power of a product rule, then apply the power of a power rule:




Apply the power of a product rule, then apply the power of a power rule:
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Which of the following expressions is equal to
?
Which of the following expressions is equal to ?
Any nonzero number raised to the power of 0 is equal to 1.
Any nonzero number raised to the power of 0 is equal to 1.
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Which quantity is greater?
(a) 
(b) 
Which quantity is greater?
(a)
(b)
(a) 
(b) 
(b) is the greater quantity.
(a)
(b)
(b) is the greater quantity.
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Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)

The two quantities are equal.
The two quantities are equal.
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is positive.
Which is the greater quantity?
(a) 
(b) 
is positive.
Which is the greater quantity?
(a)
(b)
Use the power of a power property:
(a) 
(b) 
Since
,
. Subsequently,
,
making (a) greater
Use the power of a power property:
(a)
(b)
Since ,
. Subsequently,
,
making (a) greater
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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

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
Column A gives simplifies to give us
, and Column B simplifies to give us
. 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.
Column A gives simplifies to give us , and Column B simplifies to give us
. 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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Column A Column B

Column A Column B
Anything raised to zero is equal to 1. Therefore, Column A has to be greater because 1 is greater than 0.
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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Which of the following expressions is equivalent to
?
Which of the following expressions is equivalent to
?
Use the difference of squares pattern as follows:



Use the difference of squares pattern as follows:
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Raise
to the fourth power and give the result in scientific notation.
Raise to the fourth power and give the result in scientific notation.
Use the properties of exponents to raise the number to the fourth power:




This is not in scientific notation, so adjust:



Use the properties of exponents to raise the number to the fourth power:
This is not in scientific notation, so adjust:
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Which expression is equal to 65,000?
Which expression is equal to 65,000?
is equal to 
Move the decimal one place to the right for each number of the exponent with a base ten.
For example,
,
, etc.
is equal to
Move the decimal one place to the right for each number of the exponent with a base ten.
For example, ,
, etc.
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44,000,000 can be written in scientific notation as
for some
.
Which is the greater quantity?
(A) 
(B) 8
44,000,000 can be written in scientific notation as for some
.
Which is the greater quantity?
(A)
(B) 8
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").

This requires a displacement of seven places, so

, and (B) is greater.
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").
This requires a displacement of seven places, so
, and (B) is greater.
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Give the result:

Give the result:
For any two numbers
,

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



For any two numbers ,
We can solve this problem most easily by taking advantage of this pattern, setting
:
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Give the result:

Give the result:
For any two numbers
,

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




For any two numbers ,
We can solve this problem most easily by taking advantage of this pattern, setting
:
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Give the cube of
in scientific notation.
Give the cube of in scientific notation.




This is not in scientific notation, so adjust:



This is not in scientific notation, so adjust:
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Evaluate
.
Evaluate .
By the Power of a Product Principle,

Also, by the Power of a Power Principle,

Combining these ideas,

By the Power of a Product Principle,
Also, by the Power of a Power Principle,
Combining these ideas,
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Subtract the numbers inside the parentheses first. This leaves you with 3, which you then raise to the 3rd power:

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

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

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

Multiply:
Use the FOIL method:






Use the FOIL method:
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Simplify:

Simplify:
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.
.

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.
.
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