Power of a Product: Number of terms

Solving the equationVariable in the exponent of the powerSame base and different indicatorUsing multiple rulesCombination of different basesVariables in the exponent of the powerCalculating powers with negative exponentsVariable in the base of the powerInverse formulaApplying the formula
Question 1

\( (3\times4\times5)^4= \)

Question 2

\( (4\times7\times3)^2= \)

Question 3

\( (7\cdot4\cdot6\cdot3)^4= \text{?} \)

Question 4

\( ((8by)^3)^y+(3^x)^a= \)

Question 5

\( \left(11\times15\times4\right)^6= \)

Examples with solutions for Power of a Product: Number of terms

Exercise #1

(3×4×5)4= (3\times4\times5)^4=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We apply it to the problem:

(3⋅4⋅5)4=34⋅44⋅54 (3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4 Therefore, the correct answer is option b.

Note:

From the formula of the power property mentioned above, we understand that it refers not only to two terms of the multiplication within parentheses, but also for multiple terms within parentheses.

Answer

34×44×54 3^4\times4^4\times5^4

Exercise #2

(4×7×3)2= (4\times7\times3)^2=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We apply it to the problem:

(4⋅7⋅3)2=42⋅72⋅32 (4\cdot7\cdot3)^2=4^2\cdot7^2\cdot3^2 Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we understand that we can apply it not only to the multiplication of two terms within parentheses, but is also for multiple terms within parentheses.

Answer

42×72×32 4^2\times7^2\times3^2

Exercise #3

(7⋅4⋅6⋅3)4=? (7\cdot4\cdot6\cdot3)^4= \text{?}

Video Solution

Step-by-Step Solution

We use the power property for an exponent that is applied to a set parentheses in which the terms are multiplied:

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We apply the law in the problem:

(7⋅4⋅6⋅3)4=74⋅44⋅64⋅34 (7\cdot4\cdot6\cdot3)^4=7^4\cdot4^4\cdot6^4\cdot3^4 When we apply the exponent to a parentheses with multiplication, we apply the exponent to each term of the multiplication separately, and we keep the multiplication between them.

Therefore, the correct answer is option a.

Answer

74⋅44⋅64⋅34 7^4\cdot4^4\cdot6^4\cdot3^4

Exercise #4

((8by)3)y+(3x)a= ((8by)^3)^y+(3^x)^a=

Video Solution

Step-by-Step Solution

(8by)3⋅y+3x⋅a \left(8by\right)^{3\cdot y}+3^{x\cdot a}

We begin by applying the following rule:

(am)n=am⋅n \left(a^m\right)^n=a^{m\cdot n}

We then open the parentheses according to the above rule.

(abc)x=ax⋅bx⋅cx \left(abc\right)^x=a^x\cdot b^x\cdot c^x

83y⋅b3y⋅y3y+3xa 8^{3y}\cdot b^{3y}\cdot y^{3y}+3^{xa}

Answer

83y×b3y×y3y+3ax 8^{3y}\times b^{3y}\times y^{3y}+3^{ax}

Exercise #5

(11×15×4)6= \left(11\times15\times4\right)^6=

Video Solution

Answer

All answers are correct

Question 1

\( \)

Insert the corresponding expression:

\( \left(2\times5\times4\right)^7= \)

Question 2

Insert the corresponding expression:

\( \left(8\times5\times2\right)^7= \)

Question 3

Insert the corresponding expression:

\( \left(3\times7\times9\right)^8= \)

Question 4

Insert the corresponding expression:

\( \left(12\times5\times4\right)^{10}= \)

Question 5

Insert the corresponding expression:

\( \left(2\times5\times6\right)^{15}= \)

Exercise #6

Insert the corresponding expression:

(2×5×4)7= \left(2\times5\times4\right)^7=

Video Solution

Answer

27×57×47 2^7\times5^7\times4^7

Exercise #7

Insert the corresponding expression:

(8×5×2)7= \left(8\times5\times2\right)^7=

Video Solution

Answer

87×57×27 8^7\times5^7\times2^7

Exercise #8

Insert the corresponding expression:

(3×7×9)8= \left(3\times7\times9\right)^8=

Video Solution

Answer

38×78×98 3^8\times7^8\times9^8

Exercise #9

Insert the corresponding expression:

(12×5×4)10= \left(12\times5\times4\right)^{10}=

Video Solution

Answer

1210×510×410 12^{10}\times5^{10}\times4^{10}

Exercise #10

Insert the corresponding expression:

(2×5×6)15= \left(2\times5\times6\right)^{15}=

Video Solution

Answer

215×515×615 2^{15}\times5^{15}\times6^{15}

Question 1

Insert the corresponding expression:

\( \left(8\times7\times3\right)^8= \)

Question 2

Expand the following equation:

\( \)\( \)\( \left(2\cdot6\cdot3\right)^9= \)

Question 3

Insert the corresponding expression:

\( \left(4\times10\times7\right)^9= \)

Question 4

Insert the corresponding expression:

\( \left(16\times2\times3\right)^{11}= \)

Question 5

Insert the corresponding expression:

\( \left(2\times6\times8\right)^4= \)

Exercise #11

Insert the corresponding expression:

(8×7×3)8= \left(8\times7\times3\right)^8=

Video Solution

Answer

All answers are correct

Exercise #12

Expand the following equation:

(2⋅6⋅3)9= \left(2\cdot6\cdot3\right)^9=

Video Solution

Answer

None of the answers are correct

Exercise #13

Insert the corresponding expression:

(4×10×7)9= \left(4\times10\times7\right)^9=

Video Solution

Answer

49×109×79 4^9\times10^9\times7^9

Exercise #14

Insert the corresponding expression:

(16×2×3)11= \left(16\times2\times3\right)^{11}=

Video Solution

Answer

1611×211×311 16^{11}\times2^{11}\times3^{11}

Exercise #15

Insert the corresponding expression:

(2×6×8)4= \left(2\times6\times8\right)^4=

Video Solution

Answer

24×64×84 2^4\times6^4\times8^4

Question 1

Insert the corresponding expression:

\( \left(9\times6\times8\right)^5= \)

Question 2

Insert the corresponding expression:

\( \)\( \left(9\times10\times7\right)^5= \)

Question 3

Insert the corresponding expression:

\( \left(11\times6\times5\right)^6= \)

Question 4

Insert the corresponding expression:

\( 9^{11}\times7^{11}\times6^{11}= \)

Question 5

Insert the corresponding expression:

\( 20^{10}\times4^{10}\times2^{10}= \)

Exercise #16

Insert the corresponding expression:

(9×6×8)5= \left(9\times6\times8\right)^5=

Video Solution

Answer

All answers are correct

Exercise #17

Insert the corresponding expression:

(9×10×7)5= \left(9\times10\times7\right)^5=

Video Solution

Answer

95×105×75 9^5\times10^5\times7^5

Exercise #18

Insert the corresponding expression:

(11×6×5)6= \left(11\times6\times5\right)^6=

Video Solution

Answer

a'+b' are correct

Exercise #19

Insert the corresponding expression:

911×711×611= 9^{11}\times7^{11}\times6^{11}=

Video Solution

Answer

All answers are correct

Exercise #20

Insert the corresponding expression:

2010×410×210= 20^{10}\times4^{10}\times2^{10}=

Video Solution

Answer

(20×4×2)10 \left(20\times4\times2\right)^{10}