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TEN FOR TEN®
EXPONENTS A
Exponents are shorthand. Can’t x3 be expressed “longhand” as x times x times x?
When we are asked to multiply the same base raised to powers (like x3 times x2), can we
“longhand” this as (x times x times x) times (x times x)? Take away the parentheses and we have
x times x times x times x times x, or x5. That’s why to multiply the same base raised to exponents
we keep the base and add the exponents.
Similarly, when we raise a base to a power and then raise that result to another power, like (y3)2,
we can “longhand” the inside of the parentheses (y times y times y). Now, squaring a number is
merely multiplying it times itself; doing so here will give us (y times y times y) times (y times y times
y), right? That’s why in this case we multiply the exponents.
1)
If z = y4 – 11.5, for what value of z is y = –2?
a) –6.7
c) 3.4
b) –2.3
d) 4.5
2)
[Grid In] If 94 = 3w, what is the value of w?
3)
If 0 < d < 1, which of the following statements must be true?
I.
d2 > d3
II.
d > 0.5d
III.
d > d3
a) I only
c) I and II only
b) II only
d) I and III only
e) 5.6
e) I, II, and III
PLEASE READ THE ANSWERS AND EXPLANATIONS FOR PROBLEMS 1 THROUGH 3 NOW
4)
If m2 – 9 = 9 – m2, what are all possible values of m?
a) 0 only
c) 9 only
b) 3 only
d) –3 and 3 only
e) –3, 0, and 3
5)
[Grid In] If g3 = –729, what is the value of 4g2?
6)
If 0 < v < 1, which of the following gives the correct ordering of √v, v, and v2?
7)
a) √v < v < v2
c) v < √v < v2
b) √v < v2 < v
d) v < v2 < √v
e) v2 < v < √v
If d and e are consecutive odd integers and d > e, which expression is equal to d2 – e2?
a) 2e + 1
c) 2e + 2
b) 4e – 2
d) 4e + 4
e) 4e + 6
EXPONENTS A
2
8)
If e = 5d3, what is the value of e when both d and g are doubled?
g
a) e is not changed
c) e is doubled
b) e is halved
d) e is tripled
e) e is multiplied
by 4
5v2 < (5v)2
9)
10)
For what value of v is the statement above false?
a) –5
c) 1/5
b) 0
d) 1
e)
For no value of v
If dv times d9 = d18 and (d3)w = d18, what is the value of v times w?
6/24/09
a) 12
c) 54
b) 24
d) 84
e) 91
TEN FOR TEN®
ANSWERS AND EXPLANATIONS
EXPONENTS A
Remember SADMEP. When we’re isolating a variable, we need to move all the numbers to the
other side of the equation (or inequality—sometimes a little trickier). The most efficient way to do
so is to employ SADMEP (PEMDAS backwards). First, we Subtract or Add; next, we Divide or
Multiply; finally, we deal with Exponents and only then remove any Parentheses.
1)
D. When we raise (-2) to the fourth power (or multiply it by itself four times), do we end up
with a negative or a positive number? Right, positive … 16. So, 16 – 11.5 = 4.5.
Remember, even exponents always result in non-negative numbers.
2)
8. If we had no other plan, couldn’t we multiply 9 by itself four times (and get 6561) and
then see how many 3’s we have to multiply together to get the same number?
Remember, there’s no partial credit on the SAT, and nobody checks your work. If we’d like
to solve this mathematically, however, we can first notice that 9 = 32, and so substitute 32
for 9: (32)4 = 3(2)4 = 38!
3)
E. For many of us, the most mysterious part of the number line is located between 0 and 1.
Pick a number for d (let’s say 0.5) and use it to test the three Roman numerals. Like all
other values between 0 and 1, 0.5 gets smaller the more you multiply it by itself (so I works);
next, every positive number gets smaller when you multiply it by 0.5 (so II works); and III,
which also works, is a replay of I.
PLEASE RETURN AND FINISH PROBLEMS 4 THROUGH 10
4)
D. Can we use SADMEP (see above) to combine like terms on the same side of the
equation? Adding m2 + 9 to both sides, we get 2m2 = 18, which means that m2 = 9, so
m = 3 or -3. Alternatively, we can sub in the answer choices. When we sub in zero, we get
-9 = 9, which is not true.
5)
324. The cube root of -729 is -9. When we square any negative number, however, it
becomes positive (so (-9)2 = 81). Multiplied by 4 ... If you answered 5184, you multiplied -9
by 4 before you squared it. Remember PEMDAS—exponents before multiplication. If the
test maker wanted you to multiply before applying the exponent, the expression would
have looked like this: (4g)2.
6)
E. As we raise positive values that are less than 1 to higher and higher powers (or multiply
them by themselves more and more times), they get smaller and smaller! (In case you
think that numbers between negative 1 and zero follow the same pattern, remember that
whenever they’re raised to even powers they become positive.) By that logic, mustn’t any
value that gets smaller when we raise it to a power get larger when we find its root? Try
taking the square root of ¼ (0.25); bigger, eh?
EXPONENTS A
ANSWERS AND EXPLANATIONS
2
7)
D. It’s hard to think that you would want to do this problem without Picking Numbers. How
about 3 for e and 5 for d? So, 52 – 32 = 25 – 9 = 16, which is 4 times 3 plus 4. Try it again
using other odd numbers. If you chose (a), which means you chose consecutive integers,
you really must slow down when you’re reading SAT problems.
8)
E. First, put d and g in parentheses. When you double each of them, does the “2” go
inside or outside the parentheses? Right, inside. Therefore, if we cube (2d), we’ll have to
cube the 2 along with the d, right? Doing so, we end up with 8d3. Doubling g gives us 2g.
When the numerator is multiplied by 8 and the denominator is multiplied by 2, essentially,
the expression gets multiplied by 8/2, or 4.
9)
B. Answer choice (e) is very tempting until you ask one question: What’s 0 times anything?
10)
C. Please review the notes at the beginning of the problem set that explain when to add
and when to multiply exponents. When we multiply the same base raised to powers, we
add the exponents. When we raise a base to a power and then raise the entire expression
to another power (by using parentheses), we multiply the power inside the parentheses by
the power outside the parentheses! So, v + 9 = 18 and 3w = 18.
6/24/09