Multiplication of Powers: Calculating powers with negative exponents

Using the commutative lawCombination of different basesFactoring Out the Greatest Common Factor (GCF)Numbers as coefficientsPresenting powers with negative exponents as fractionsSolving the problemVariable in the exponent of the powerIdentify the greater valueVariables in the exponent of the powerInverse formulaVariable in the base of the powerNumber of termsSystem of equations with no solutionApplying the formulaUsing multiple rules
Question 1

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

Question 2

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

Question 3

\( y^{-2}\times y^7= \)

Question 4

Reduce the following equation:

\( 2^4\times2^{-2}\times2^3= \)

Question 5

Simplify the following equation:

\( 4^{-2}\times4^{-4}= \)

Examples with solutions for Multiplication of Powers: Calculating powers with negative exponents

Exercise #1

75⋅7−6=? 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):

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

75⋅7−6=75+(−6)=75−6=7−1 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:

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

7−1=171=17 7^{-1}=\frac{1}{7^1}=\frac{1}{7} We then summarise the solution to the problem: 75⋅7−6=7−1=17 7^5\cdot7^{-6}=7^{-1}=\frac{1}{7} Therefore, the correct answer is option B.

Answer

17 \frac{1}{7}

Exercise #2

124⋅12−6=? 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:

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

124⋅12−6=124+(−6)=124−6=12−2 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:

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

12−2=1122=1144 12^{-2}=\frac{1}{12^2}=\frac{1}{144} Lastly we summarise the solution to the problem: 124⋅12−6=12−2=1144 12^4\cdot12^{-6}=12^{-2} =\frac{1}{144} Therefore, the correct answer is option A.

Answer

1144 \frac{1}{144}

Exercise #3

y−2×y7= y^{-2}\times y^7=

Video Solution

Answer

y5 y^5

Exercise #4

Reduce the following equation:

24×2−2×23= 2^4\times2^{-2}\times2^3=

Video Solution

Answer

24−2+3 2^{4-2+3}

Exercise #5

Simplify the following equation:

4−2×4−4= 4^{-2}\times4^{-4}=

Video Solution

Answer

4−2−4 4^{-2-4}

Question 1

Simplify the following equation:

\( 3^{-4}\times3^{-2}= \)

Question 2

Simplify the following equation:

\( 2^6\times2^{-3}= \)

Question 3

Insert the corresponding expression:

\( 8^4\times8\times8^{-1}= \)

Question 4

Insert the corresponding expression:

\( 7^{-2}\times7^{-3}\times7^5= \)

Question 5

Reduce the following equation:

\( 5^{-2}\times5^{-1}\times5= \)

Exercise #6

Simplify the following equation:

3−4×3−2= 3^{-4}\times3^{-2}=

Video Solution

Answer

3−4−2 3^{-4-2}

Exercise #7

Simplify the following equation:

26×2−3= 2^6\times2^{-3}=

Video Solution

Answer

26−3 2^{6-3}

Exercise #8

Insert the corresponding expression:

84×8×8−1= 8^4\times8\times8^{-1}=

Video Solution

Answer

84+1−1 8^{4+1-1}

Exercise #9

Insert the corresponding expression:

7−2×7−3×75= 7^{-2}\times7^{-3}\times7^5=

Video Solution

Answer

7−2−3+5 7^{-2-3+5}

Exercise #10

Reduce the following equation:

5−2×5−1×5= 5^{-2}\times5^{-1}\times5=

Video Solution

Answer

5−2 5^{-2}

Question 1

Reduce the following equation:

\( 9^{-3}\times9^{-5}\times9^{-2}= \)

Question 2

Reduce the following equation:

\( \)\( 10\times10^{-3}\times10^5= \)

Question 3

Reduce the following equation:

\( 8^{-10}\times8^5\times8^4= \)

Question 4

Reduce the following equation:

\( 3^{-2}\times3^4= \)

Question 5

Reduce the following equation:

\( 5^{-3}\times5^{-4}= \)

Exercise #11

Reduce the following equation:

9−3×9−5×9−2= 9^{-3}\times9^{-5}\times9^{-2}=

Video Solution

Answer

9−10 9^{-10}

Exercise #12

Reduce the following equation:

10×10−3×105= 10\times10^{-3}\times10^5=

Video Solution

Answer

103 10^3

Exercise #13

Reduce the following equation:

8−10×85×84= 8^{-10}\times8^5\times8^4=

Video Solution

Answer

8−1 8^{-1}

Exercise #14

Reduce the following equation:

3−2×34= 3^{-2}\times3^4=

Video Solution

Answer

32 3^2

Exercise #15

Reduce the following equation:

5−3×5−4= 5^{-3}\times5^{-4}=

Video Solution

Answer

5−7 5^{-7}

Question 1

Reduce the following equation:

\( 6^{-7}\times6^3= \)

Question 2

Insert the corresponding expression:

\( \)\( 6^{-5}\times6^2= \)

Question 3

Reduce the following equation:

\( 11^{-2}\times11^{-5}\times11^{-4}= \)

Question 4

Insert the corresponding expression:

\( 9^{-1}\times9^{-2}\times9^{-3}= \)

Question 5

Insert the corresponding expression:

\( 8^{-10}\times8^{-5}\times8^9= \)

Exercise #16

Reduce the following equation:

6−7×63= 6^{-7}\times6^3=

Video Solution

Answer

6−4 6^{-4}

Exercise #17

Insert the corresponding expression:

6−5×62= 6^{-5}\times6^2=

Video Solution

Answer

163 \frac{1}{6^3}

Exercise #18

Reduce the following equation:

11−2×11−5×11−4= 11^{-2}\times11^{-5}\times11^{-4}=

Video Solution

Answer

11111 \frac{1}{11^{11}}

Exercise #19

Insert the corresponding expression:

9−1×9−2×9−3= 9^{-1}\times9^{-2}\times9^{-3}=

Video Solution

Answer

All answers are correct

Exercise #20

Insert the corresponding expression:

8−10×8−5×89= 8^{-10}\times8^{-5}\times8^9=

Video Solution

Answer

186 \frac{1}{8^6}