Exponential Ratios and Rational Numbers - PSAT Math
Card 0 of 133
Solve for
.

Solve for .
Since 
Hence 
Since
Hence
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If
and
are positive integers and
, then what is the value of
?
If and
are positive integers and
, then what is the value of
?
43 = 64
Alternatively written, this is 4(4)(4) = 64 or 43 = 641.
Thus, m = 3 and n = 1.
m/n = 3/1 = 3.
43 = 64
Alternatively written, this is 4(4)(4) = 64 or 43 = 641.
Thus, m = 3 and n = 1.
m/n = 3/1 = 3.
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Write the following logarithm in expanded form:

Write the following logarithm in expanded form:
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From the equation in the problem statement

Now squaring both sides we get
this is a quadratic equation which equals

and the factors of this equation are

This gives us
.
However, if we plug these solutions back into the original equation,
does not create an equality. Therefore,
is an extraneous solution.
From the equation in the problem statement
Now squaring both sides we get
this is a quadratic equation which equals
and the factors of this equation are
This gives us .
However, if we plug these solutions back into the original equation, does not create an equality. Therefore,
is an extraneous solution.
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Rationalize the denominator:

Rationalize the denominator:
The conjugate of
is
.
Now multiply both the numerator and the denominator by 
and you get:


Hence we get

The conjugate of is
.
Now multiply both the numerator and the denominator by
and you get:
Hence we get
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Solve for
:

Solve for :
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Solve for
.

Solve for .
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If,

What does 
If,
What does
If
,
then
.
If ,
then .
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For some positive integer
, if
, what is the value of
?
For some positive integer , if
, what is the value of
?
If
, then
must equal 3 (Note that
cannot be -3 because you need it to be positive.
Now, plug
into the new equation
:



If , then
must equal 3 (Note that
cannot be -3 because you need it to be positive.
Now, plug into the new equation
:
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If a piece of pie is cut into 3 sections, and each of those pieces is further cut into three sections, then those pieces are cut into three sections, how many (tiny) pieces of pie are there?
If a piece of pie is cut into 3 sections, and each of those pieces is further cut into three sections, then those pieces are cut into three sections, how many (tiny) pieces of pie are there?
The answer is 33 = 27
The answer is 33 = 27
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Which of the following lists the above quantities from least to greatest?
Which of the following lists the above quantities from least to greatest?
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Simplify:
![\sqrt[3]{x^{10}y^{8}z^{9}}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/25511/gif.latex)
Simplify:
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Solve for
:

Solve for :
From the equation one can see that

Hence
must be equal to 25.
From the equation one can see that
Hence must be equal to 25.
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Evaluate:

Evaluate:
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Solve for
.

Solve for .
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Solve for
.

Solve for .
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Solve for
.

Solve for .
Use the rules of logarithms to combine terms.

Hence, 


By fatoring we get 
Hence
.
However, you cannot take the logarithm of a negative number. Thus, the only value for
is
.
Use the rules of logarithms to combine terms.
Hence,
By fatoring we get
Hence .
However, you cannot take the logarithm of a negative number. Thus, the only value for is
.
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and

Find 
and
Find

Hence the correct answer is
.
Hence the correct answer is .
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Which of the following lists the above quantities from least to greatest?
Which of the following lists the above quantities from least to greatest?
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