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Example Questions
Example Question #1 : How To Divide Exponents
For all real numbers n, (2^{n} * 2) / (2^{n} * 2^{n}) =
2n – 1
2n
2n/2n
2^{1 – n}
2
2^{1 – n}
(2^{n} * 2) / (2^{n} * 2^{n}) simplifies to 2/2^{n} or 2^{1}/2^{n}.
When dividing exponents with the same base, you subtract the divisor from the dividend, giving 2^{1–n}.
Example Question #2 : How To Divide Exponents
If x^{9}/x^{3} = x^{n}, solve for n.
7
3
6
12
9
6
When dividing terms with the same base, we can subtract the exponents:
9 – 3 = 6
Example Question #3 : How To Divide Exponents
Simplify the following expression: (x^{2}y^{4})/(x^{3}y^{3}z^{2})
y/xz^{2}
xz^{2}/y
xy/z^{2}
z^{2}xy
y/xz^{2}
According to the rules of exponents, a^{x}/a^{y}^{ } = a^{xy}
In this expression, we can follow this rule to simplify x^{2}/x^{3 }and y^{4}/y^{3}
x^{2–3} = x^{–1} = 1/x. y^{4–3} = y^{1 }= y.
Therefore, y/xz^{2}
Example Question #4 : How To Divide Exponents
Simplify:
x^{9 }/ z^{9}
(z/x)^{3 }/ 2
x^{3}z^{9 }/ y^{9}
x^{4 }/ z^{4}
x^{6}y^{6}z^{6}
x^{9 }/ z^{9}
When dividing, subtract exponents (x^{a}/x^{b} = x^{(a – b)}.) Therefore, the quantity in the parenthesis is: x^{(4 – (–2))} * y^{(–3 – (–3))} * z^{(–1 – 5)} = x^{6}/z^{6}. Raising ^{ }this to the 3/2 power results in multiplying the exponents by 3/2: x^{6 * 3/2}/z^{6 * 3/2} = x^{9}/z^{9}.
Example Question #5 : How To Divide Exponents
Half of the radioactive nuclei of a substance decays in a week. If a sample started with 10^{10} nuclei, how many have decayed after 28 days?
6.25 x 108
106
28 x 1010
9.375 x 109
1010
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 10^{10} = 9. 375 x 10^{9}
Example Question #6 : How To Divide Exponents

5. Simplify the problem (x^{4}y^{2}/x^{5})^{3}
x^{3}y^{6}
x^{4}/y^{6}
x/y
y^{5}/x
y^{6}/x^{3}
y^{6}/x^{3}
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 (x^{4}/x^{5}) = x^{–1 }= 1/x; Still using order of operations (PEMDAS) we get the following:(x^{4}y^{2}/x^{5})^{3}= (y^{2}/x)^{3} = y^{6}/(x^{3}).
Example Question #27 : Exponential Operations
If x^{7} / x^{3/2} = x^{n}, what is the value of n?
21/2
21/2
10/2
17/2
11/2
17/2
x^{7} / x^{3/2} = x^{7} (x^{+3/2}) based on the fact that division changes the sign of an exponent.
x^{7} (x^{+3/2}) = x^{7}^{+3/2 }due to the additive property of exponent numbers that are multiplied.
7+3/2= 14/2 + 3/2 = 17/2 so^{}
x^{7} / x^{3/2} = x^{7}^{+3/2 }= x^{17/2}
Since x^{7} / x^{3/2} = x^{n}, x^{n}^{ }= x^{17/2}
So n = 17/2
Example Question #28 : Exponential Operations
Simplify x^{2}x^{4}y/y^{2}x
y/x^{5}
x^{7}y^{2}
x^{5}/y
7xy^{2}
x^{5}/y^{3}
x^{5}/y
1) According to the rules of exponents, one can add the exponents when adding to variables with the same base. So, x^{2}x^{4} becomes x^{6}.
2) The rules of exponents also state that if the bases are the same, one can substract the exponents when dividing. So, x^{6}/x becomes x^{5}. Similarly, y/y^{2 }becomes 1/y.
3) When combining these operations, one gets x^{5}/y.
Example Question #7 : How To Divide Exponents
a^{3}(b^{3}√b)(c)
^{}a^{2}b^{3}√c
(b^{7})/(a^{3}c)
(b^{3}√b)/(a^{3}c)
(b^{3})/(a^{3}c^{2})
(b^{3}√b)/(a^{3}c)
Example Question #8 : How To Divide Exponents
5^{4 }/ 25 =
10
25
50
5^{4} / 5
5
25
25 = 5 * 5 = 5^{2}. Then 5^{4} / 25 = 5^{4} / 5^{2}.
Now we can subtract the exponents because the operation is division. 5^{4} / 5^{2 }= 5^{4 – 2} = 5^{2} = 25. The answer is therefore 25.