PSAT Math : Exponents
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Example Question #21 : Exponents
For all real numbers n, (2n * 2) / (2n * 2n) =
2n/2n
21 – n
2
2n – 1
2n
21 – n
(2n * 2) / (2n * 2n) simplifies to 2/2n or 21/2n.
When dividing exponents with the same base, you subtract the divisor from the dividend, giving 21–n.
Example Question #1 : How To Divide Exponents
If x9/x3 = xn, solve for n.
3
7
6
9
12
6
When dividing terms with the same base, we can subtract the exponents:
9 – 3 = 6
Example Question #1 : How To Divide Exponents
Simplify the following expression: (x2y4)/(x3y3z2)
xy/z2
xz2/y
z2xy
y/xz2
y/xz2
According to the rules of exponents, ax/ay = ax-y
In this expression, we can follow this rule to simplify x2/x3 and y4/y3
x2–3 = x–1 = 1/x. y4–3 = y1 = y.
Therefore, y/xz2
Example Question #22 : Exponents
Simplify:
x4 / z4
x3z9 / y9
x9 / z9
(z/x)3 / 2
x6y6z6
x9 / z9
When dividing, subtract exponents (xa/xb = x(a – b).) Therefore, the quantity in the parenthesis is: x(4 – (–2)) * y(–3 – (–3)) * z(–1 – 5) = x6/z6. Raising this to the 3/2 power results in multiplying the exponents by 3/2: x6 * 3/2/z6 * 3/2 = x9/z9.
Example Question #23 : Exponents
Half of the radioactive nuclei of a substance decays in a week. If a sample started with 1010 nuclei, how many have decayed after 28 days?
28 x 1010
1010
9.375 x 109
6.25 x 108
106
9.375 x 109
If half of the sample decays each week: 1/2 is left after one week, 1/4 is left after two weeks, 1/8 is left after three weeks and 1/16 is left after four weeks (28 days.) That means that 15/16 has decayed. 15/16 x 1010 = 9. 375 x 109
Example Question #24 : Exponents
-
5. Simplify the problem (x4y2/x5)3
x4/y6
y6/x3
y5/x
x3y6
x/y
y6/x3
Properties of exponents suggests that when multiplying the same base, add the exponents, when dividing, subtract the exponents on bottom from those on top, and when raising an exponent to another power, multiply the exponents. Remember that (x4/x5) = x–1 = 1/x; Still using order of operations (PEMDAS) we get the following:(x4y2/x5)3= (y2/x)3 = y6/(x3).
Example Question #25 : Exponents
If x7 / x-3/2 = xn, what is the value of n?
17/2
11/2
21/2
-21/2
10/2
17/2
x7 / x-3/2 = x7 (x+3/2) based on the fact that division changes the sign of an exponent.
x7 (x+3/2) = x7+3/2 due to the additive property of exponent numbers that are multiplied.
7+3/2= 14/2 + 3/2 = 17/2 so
x7 / x-3/2 = x7+3/2 = x17/2
Since x7 / x-3/2 = xn, xn = x17/2
So n = 17/2
Example Question #26 : Exponents
Simplify x2x4y/y2x
x5/y3
y/x5
7xy2
x5/y
x7y2
x5/y
1) According to the rules of exponents, one can add the exponents when adding to variables with the same base. So, x2x4 becomes x6.
2) The rules of exponents also state that if the bases are the same, one can substract the exponents when dividing. So, x6/x becomes x5. Similarly, y/y2 becomes 1/y.
3) When combining these operations, one gets x5/y.
Example Question #27 : Exponents
(b3√b)/(a3c)
(b7)/(a3c)
(b3)/(a3c2)
a3(b3√b)(c)
a2b3√c
(b3√b)/(a3c)
Example Question #28 : Exponents
54 / 25 =
25
10
5
54 / 5
50
25
25 = 5 * 5 = 52. Then 54 / 25 = 54 / 52.
Now we can subtract the exponents because the operation is division. 54 / 52 = 54 – 2 = 52 = 25. The answer is therefore 25.
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