How to find the exponent of variables - ISEE Upper Level Quantitative Reasoning
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Simplify:

Simplify:
Apply the power of a product property:





Apply the power of a product property:
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What is the coefficient of
in the expansion of
.
What is the coefficient of in the expansion of
.
By the Binomial Theorem, if
is expanded, the coefficient of
is
.
Substitute
: The coefficient of
is:



By the Binomial Theorem, if is expanded, the coefficient of
is
.
Substitute : The coefficient of
is:
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What is the coefficient of
in the expansion of
?
What is the coefficient of in the expansion of
?
By the Binomial Theorem, the
term of
is
.
Substitute
and this becomes
.
The coefficient is
.
By the Binomial Theorem, the term of
is
.
Substitute and this becomes
.
The coefficient is
.
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What is the coefficient of
in the expansion of
?
What is the coefficient of in the expansion of
?
By the Binomial Theorem, the
term of
is
,
making the coefficient of 
.
We can set
in this expression:




By the Binomial Theorem, the term of
is
,
making the coefficient of
.
We can set in this expression:
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Simplify the expression: ![\left [\left ( x ^{3} \right )^{3} \right ]^{3 }](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/148608/gif.latex)
Simplify the expression:
Apply the power of a power property twice:
![\left [\left ( x ^{3} \right )^{3} \right ]^{3 } = ( x ^{3; \cdot ; 3 } )^{3 } = ( x ^{9 } )^{3 } = x ^{9\cdot 3} =x ^{27}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/148609/gif.latex)
Apply the power of a power property twice:
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Evaluate:
![\left [ (x^4)^4 \right ]^5](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/152827/gif.latex)
Evaluate:
We need to apply the power of power rule twice:
![\left [ (x^4)^4 \right ]^5=(x^{4\times 4})^5=(x^{16})^5=x^{16\times 5}=x^{80}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/152828/gif.latex)
We need to apply the power of power rule twice:
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Solve for
.
Solve for .
Based on the power of a product rule we have:

The bases are the same, so we can write:

Based on the power of a product rule we have:
The bases are the same, so we can write:
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Simplify:

Simplify:
First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

Apply the exponent within the parentheses and simplify.




This fraction cannot be simplified further.
First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.
Apply the exponent within the parentheses and simplify.
This fraction cannot be simplified further.
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Simplify:

Simplify:
First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

Apply the exponent within the parentheses and simplify.




First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.
Apply the exponent within the parentheses and simplify.
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Simplify if
and
.

Simplify if and
.
Begin by factoring the numerator and denominator.
can be factored out of each term.

can be canceled, since it appears in both the numerator and denomintor.

Next, factor the numerator.

Simplify.

Begin by factoring the numerator and denominator. can be factored out of each term.
can be canceled, since it appears in both the numerator and denomintor.
Next, factor the numerator.
Simplify.
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Evaluate
.
Evaluate .
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Evaluate
.
Evaluate .
To solve for the variable isolate it on one side of the equation with all of constants on the other side.

First add one third to both sides.


Calculate a common denominator to add the two fractions.




Square both sides to solve for y.

To solve for the variable isolate it on one side of the equation with all of constants on the other side.
First add one third to both sides.
Calculate a common denominator to add the two fractions.
Square both sides to solve for y.
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Evaluate
.
Evaluate .
By the Power of a Product Principle,

Also, by the Power of a Power Principle,

Combining these ideas, then substituting:






By the Power of a Product Principle,
Also, by the Power of a Power Principle,
Combining these ideas, then substituting:
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Evaluate
.
Evaluate .
By the Power of a Power Principle,

So

Also, by the Power of a Product Principle,

, so, substituting,
.
By the Power of a Power Principle,
So
Also, by the Power of a Product Principle,
, so, substituting,
.
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Simplify the following:

Simplify the following:
Simplify the following:

Let's recall the rules for distributing exponents.
We treat coefficients (like the 7) like regular numbers and raise them to the new exponent.
We deal with variables (like the t, h, and b) by multiplying their current exponent by the new exponent.
Doing so yields:

Simplify to get:

Simplify the following:
Let's recall the rules for distributing exponents.
We treat coefficients (like the 7) like regular numbers and raise them to the new exponent.
We deal with variables (like the t, h, and b) by multiplying their current exponent by the new exponent.
Doing so yields:
Simplify to get:
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Simplify:

Simplify:
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Which is greater?
(a) 
(b) 
Which is greater?
(a)
(b)
If
, then
and 
, so by transitivity,
, and (b) is greater
If , then
and
, so by transitivity,
, and (b) is greater
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Which is greater?
(a) 
(b) 
Which is greater?
(a)
(b)

A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.
A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.
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Expand: 
Which is the greater quantity?
(a) The coefficient of 
(b) The coefficient of 
Expand:
Which is the greater quantity?
(a) The coefficient of
(b) The coefficient of
By the Binomial Theorem, if
is expanded, the coefficient of
is
.
(a) Substitute
: The coerfficient of
is
.
(b) Substitute
: The coerfficient of
is
.
The two are equal.
By the Binomial Theorem, if is expanded, the coefficient of
is
.
(a) Substitute : The coerfficient of
is
.
(b) Substitute : The coerfficient of
is
.
The two are equal.
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Expand: 
Which is the greater quantity?
(a) The coefficient of 
(b) The coefficient of 
Expand:
Which is the greater quantity?
(a) The coefficient of
(b) The coefficient of
Using the Binomial Theorem, if
is expanded, the
term is

.
This makes
the coefficient of
.
We compare the values of this expression at
for both
and
.
(a) If
and
, the coefficient is
.
This is the coefficient of
.
(b) If
and
, the coefficient is
.
This is the coefficient of
.
(b) is the greater quantity.
Using the Binomial Theorem, if is expanded, the
term is
.
This makes the coefficient of
.
We compare the values of this expression at for both
and
.
(a) If and
, the coefficient is
.
This is the coefficient of .
(b) If and
, the coefficient is
.
This is the coefficient of .
(b) is the greater quantity.
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