Examples with solutions for Negative Exponents: Multiplying Exponents with the same base

Exercise #1

7576=? 7^5\cdot7^{-6}=\text{?}

Video Solution

Step-by-Step Solution

We begin by using the rule for multiplying exponents. (the multiplication between terms with identical bases):

aman=am+n a^m\cdot a^n=a^{m+n} We then apply it to the problem:

7576=75+(6)=756=71 7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1} When in a first stage we begin by applying the aforementioned rule and then continue on to simplify the expression in the exponent,

Next, we use the negative exponent rule:

an=1an a^{-n}=\frac{1}{a^n} We apply it to the expression obtained in the previous step:

71=171=17 7^{-1}=\frac{1}{7^1}=\frac{1}{7} We then summarise the solution to the problem: 7576=71=17 7^5\cdot7^{-6}=7^{-1}=\frac{1}{7} Therefore, the correct answer is option B.

Answer

17 \frac{1}{7}

Exercise #2

124126=? 12^4\cdot12^{-6}=\text{?}

Video Solution

Step-by-Step Solution

We begin by using the power rule of exponents; for the multiplication of terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} We apply it to the given problem:

124126=124+(6)=1246=122 12^4\cdot12^{-6}=12^{4+(-6)}=12^{4-6}=12^{-2} When in a first stage we apply the aforementioned rule and then simplify the subsequent expression in the exponent,

Next, we use the negative exponent rule:

an=1an a^{-n}=\frac{1}{a^n} We apply it to the expression that we obtained in the previous step:

122=1122=1144 12^{-2}=\frac{1}{12^2}=\frac{1}{144} Lastly we summarise the solution to the problem: 124126=122=1144 12^4\cdot12^{-6}=12^{-2} =\frac{1}{144} Therefore, the correct answer is option A.

Answer

1144 \frac{1}{144}

Exercise #3

9300192529549=? 9^{300}\cdot\frac{1}{9^{-252}}\cdot9^{-549}=\text{?}

Video Solution

Answer

193 \frac{1}{9^{-3}}

Exercise #4

(18)8(18)3=? (-\frac{1}{8})^8\cdot(-\frac{1}{8})^{-3}=?

Video Solution

Answer

85 -8^{-5}

Exercise #5

42x1442=? 4^{2x}\cdot\frac{1}{4}\cdot4^{-2}=\text{?}

Video Solution

Answer

1432x \frac{1}{4^{3-2x}}

Exercise #6

133453=? \frac{1}{-3}\cdot3^{-4}\cdot5^3=\text{?}

Video Solution

Answer

5335 -\frac{5^3}{3^5}