Tip: Know the Rules of Exponents. a4=a2+2=a2×a2=10×10=100.
Solution 2:
Tip: Know the Rules of Exponents.
Since a2=10, so a=10. Then, a4=(10)4=(1021)4=10(21×4)=102=100.
Incorrect Choices:
(A) Tip: The simplest choice may not be the correct one.
(B) Tip: Read the entire question carefully.
If you solve for a, you will obtain a=10. However, the question asks for a4.
(C) Tip: Just because a number appears in the question doesn't mean it is the answer.
(E) Tip: Read the entire question carefully.
If you forget the exponent and think that a1=10, you will obtain a4=104=10000.
If a2x⋅a3=a15 and 3by=b5, what is the value of y−x?
(A) −1
(B) 6
(C) 9
(D) 225
(E) 21
Correct Answer: C
Solution:
Tip: Know the Rules of Exponents.
Using the first equation, we solve for x.
a2x⋅a3a2x+32x+32xx=====a15a1515126first equationapplyam⋅an=am+nto left sideequate exponentssubtract3from both sidesdivide both sides by2
Similarly, using the second equation, we solve for y.
3byb3y3yy====b5b5515second equationapplynam=anmto left sideequate exponentsmultiply both sides by3
Now that we know x=6 and y=15, we find y−x=15−6=9.
Incorrect Choices:
(A) Tip: Read the entire question carefully.
If you ignore the cube root in the second equation and solve by=b5 instead of 3by=b5, you will get y=5. Then y−x=5−6=−1. But this is wrong.
(B) Tip: Read the entire question carefully.
If you solve for x, instead of y−x, you will get this wrong answer.
(D) Tip: Know the Rules of Exponents.
If you apply the wrong formula am⋅an=am⋅n to the first equation, like this:
a2x⋅a3a2x⋅3==a15a15first equationmistake: multiplied exponents, instead of added
you will get x=25. Then y−x=15−25=225. The correct rule states am⋅an=am+n.
(E) Tip: Read the entire question carefully.
If you solve for y+x instead of y−x, you will get this wrong answer.
A
B
C
D
E
If a and b are positive non-consecutive integers and 4a⋅4b=1024, which of the following equals 2a+2b?
(A) 5
(B) 12
(C) 18
(D) 32
(E) 1024
The correct answer is: C
The expression (y4)3(3y3)(2y2)2⋅(y0)3 is equivalent to which one of the following?
(A) 12y−5
(B) 6y−5
(C) 12y−2
(D) 12
(E) 12y5
Correct Answer: A
Solution:
Tip: Know the Rules of Exponents.
Using the rules of exponents, we simplify the given expression.
(B) Tip: Know the Rules of Exponents.
If in step (3) you don't apply the rule (a ⋅b)n=an⋅bn correctly, you will get this wrong answer. That is,
(y4)3(3y3)(2y2⋅2)mistake: did not raise2to the power of2(3)
(C) Tip: Read the entire question carefully.
If you forget the 0 in the multiplicand's exponent and read (y4)3(3y3)(2y2)2⋅(y)3 instead of (y4)3(3y3)(2y2)2⋅(y0)3, you will get this wrong answer.
(D) Tip: Know the Rules of Exponents.
If in step (7) you apply the wrong rule (an)m=an + m to the denominator, as shown below, you will get this wrong answer. Don't add the exponents when you should be multiplying them.
y4+312y7mistake: added exponentsinstead of multiplied(7)
The correct rule states (an)m=an⋅m.
(E) Tip: Know the Rules of Exponents. Tip: Be careful with signs!
If in the step (9) you apply the wrong rule anam=an-m, you will get this wrong answer. The correct rule states anam=am−n.
Alternatively, you may not notice or you may forget that the exponent becomes negative. Be careful!
Review
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