When finding an expression with multiplication or an exercise that has only multiplication operations inside a parenthesis and the wholes expression is raised to a certain exponent, we can take the exponent and apply it to each of the terms of the expression or exercise.
We must not forget to keep the multiplication signs between the terms.
Property formula:
(a×b)n=an×bn (a\times b)^n=a^n\times b^n
This property also pertains to algebraic expressions.

Suggested Topics to Practice in Advance

  1. Multiplying Exponents with the Same Base
  2. Division of Exponents with the Same Base

Practice Power of a Product

Examples with solutions for Power of a Product

Exercise #1

Insert the corresponding expression:

(2×11)5= \left(2\times11\right)^5=

Video Solution

Step-by-Step Solution

To solve the expression (2×11)5 \left(2\times11\right)^5 , we can apply the rule for the power of a product, which states that(a×b)n=an×bn \left(a \times b\right)^n = a^n \times b^n .

In this case, our expression is (2×11)5 \left(2\times11\right)^5 , wherea=2 a = 2 and b=11 b = 11 , and n=5 n = 5 .

Applying the power of a product rule gives us:

  • an=25 a^n = 2^5

  • bn=115 b^n = 11^5

Therefore, (2×11)5=25×115 \left(2\times11\right)^5 = 2^5 \times 11^5 .

Answer

25×115 2^5\times11^5

Exercise #2

Insert the corresponding expression:

(10×3)4= \left(10\times3\right)^4=

Video Solution

Step-by-Step Solution

To solve this problem, we'll apply the power of a product rule to the expression (10×3)4(10 \times 3)^4.

  • Step 1: Identify the expression.
    The given expression is (10×3)4(10 \times 3)^4.

  • Step 2: Apply the power of a product law.
    According to the rule, (a×b)n=an×bn(a \times b)^n = a^n \times b^n, our expression becomes:
    (10×3)4=104×34(10 \times 3)^4 = 10^4 \times 3^4.

  • Step 3: Evaluate the choices:
    - First choice: 34×1043^4 \times 10^4
    Rearranging terms, this is equivalent to 104×3410^4 \times 3^4. Therefore, it matches our transformed expression.
    - Second choice: 30430^4
    Since 30=10×330 = 10 \times 3, (10×3)4=304(10 \times 3)^4 = 30^4. This simplifies to the same expression.
    - Third choice: 104×3410^4 \times 3^4
    This is exactly what we found using the power of a product rule

Thus, the solution is that all answers are correct.

Answer

All answers are correct

Exercise #3

Insert the corresponding expression:

(9×7)4= \left(9\times7\right)^4=

Video Solution

Step-by-Step Solution

To solve the problem, we need to apply the exponent rule known as the Power of a Product, which states that if you have a product raised to an exponent, you can apply the exponent to each factor in the product individually.

The general form of this rule is:

(a×b)n=an×bn (a \times b)^n = a^n \times b^n

According to this formula, when we have the expression:

(9×7)4 (9 \times 7)^4

We apply the exponent 4 to each factor within the parentheses. This process results in:

  • Raising 9 to the power of 4: 94 9^4
  • Raising 7 to the power of 4: 74 7^4

Therefore, the expression simplifies to:

94×74 9^4 \times 7^4

This demonstrates the application of the Power of a Product rule successfully, showing detailed steps and the correct application of exponential laws.

Answer

94×74 9^4\times7^4

Exercise #4

Insert the corresponding expression:

(6×8)4= \left(6\times8\right)^4=

Video Solution

Step-by-Step Solution

To solve the problem of rewriting the expression (6×8)4(6 \times 8)^4, we will apply the Power of a Product rule. This rule states that (a×b)n=an×bn(a \times b)^n = a^n \times b^n.

Let's apply this rule to the given expression:

(6×8)4(6 \times 8)^4 can be rewritten as 64×846^4 \times 8^4 by applying the Power of a Product rule.

Now, let's consider the choices given:

  • Choice 1: 64×846^4 \times 8^4 - This choice directly corresponds to our rewritten expression using the Power of a Product rule.

  • Choice 2: 48448^4 - This represents the product calculated first (6×8=486 \times 8 = 48) and then raised to the fourth power. This is also a valid representation since (6×8)4(6 \times 8)^4 is equivalent to 48448^4.

  • Choice 3: 64×86^4 \times 8 - This is incorrect because it does not correctly apply the Power of a Product rule.

Therefore, the correct answer according to the given choices is "a' + b' are correct," which indicates the validity of both 64×846^4 \times 8^4 and 48448^4 as representations of (6×8)4(6 \times 8)^4.

Answer

a'+b' are correct

Exercise #5

Insert the corresponding expression:

(5×7)3= \left(5\times7\right)^3=

Video Solution

Step-by-Step Solution

The problem requires us to simplify the expression (5×7)3(5 \times 7)^3 using the power of a product rule.

The power of a product rule states that for any numbers a a and b b , and any integer n n , the expression (a×b)n (a \times b)^n can be expanded to an×bn a^n \times b^n .

Applying this rule to the given expression:

  • Identify the values of a a and b b as 5 5 and 7 7 , respectively.

  • Identify n n as 3 3 .

  • Substitute into the rule:
    (5×7)3=53×73(5 \times 7)^3 = 5^3 \times 7^3

The simplified expression is therefore: 53×735^3 \times 7^3.

Answer

53×73 5^3\times7^3

Exercise #6

Insert the corresponding expression:

(2×6)3= \left(2\times6\right)^3=

Video Solution

Step-by-Step Solution

We are given the expression (2×6)3 \left(2\times6\right)^3 and need to simplify it using the power of a product rule in exponents.

The power of a product rule states that when you have a product inside a power, you can apply the exponent to each factor in the product individually. In mathematical terms, the rule is expressed as:

  • (ab)n=anbn (a \cdot b)^n = a^n \cdot b^n

Applying this to our expression, we have:

(2×6)3=23×63 \left(2\times6\right)^3 = 2^3\times6^3

This means that each term inside the parentheses is raised to the power of 3 separately.

Therefore, the expression (2×6)3 \left(2\times6\right)^3 simplifies to 23×63 2^3\times6^3 as per the power of a product rule.

Answer

23×63 2^3\times6^3

Exercise #7

Insert the corresponding expression:

(5×3)3= \left(5\times3\right)^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll clarify our understanding and execution as follows:

  • Identify the given expression: (5×3)3 (5 \times 3)^3 .

  • Apply the relevant formula: The power of a product rule is given by (a×b)n=an×bn (a \times b)^n = a^n \times b^n .

  • Execution: Rewrite (5×3)3 (5 \times 3)^3 as 53×33 5^3 \times 3^3 .

Let's walk through the solution:

Initially, we have the expression (5×3)3 (5 \times 3)^3 . We recognize that by the power of a product rule, this can be rewritten as 53×33 5^3 \times 3^3 .

Next, let's verify the choices:
- 53×3 5^3 \times 3 is incorrect as the exponent doesn't apply to both factors.
- 153 15^3 represents the base simplified (5×3=15 5\times3=15 ).
- 53×33 5^3 \times 3^3 matches exactly what we derived earlier, making this the correct expression resulting from the power rule.

Therefore, the correct choice is option "B+C are correct". This acknowledges that "B" expresses 15315^3, and "C" independently describes the valid expanded expression, both right solutions for different cases of interpretations.

Answer

B+C are correct

Exercise #8

Insert the corresponding expression:

(4×2)2= \left(4\times2\right)^2=

Video Solution

Step-by-Step Solution

To solve the problem (4×2)2 \left(4\times2\right)^2, we need to apply the rule of exponents known as the "Power of a Product". This rule states that (ab)n=an×bn (ab)^n = a^n \times b^n .

Here, a=4 a = 4 , b=2 b = 2 , and n=2 n = 2 .

  • Step 1: Apply the "Power of a Product" rule: (4×2)2=42×22 \left(4 \times 2\right)^2 = 4^2 \times 2^2 .

Thus, the expression (4×2)2 \left(4\times2\right)^2 is equivalent to 42×22 4^2 \times 2^2 .

Answer

42×22 4^2\times2^2

Exercise #9

Insert the corresponding expression:

(2×3)2= \left(2\times3\right)^2=

Video Solution

Step-by-Step Solution

The given expression is (2×3)2 \left(2\times3\right)^2. We need to apply the rule of exponents known as the "Power of a Product." This rule states that when you have a product raised to an exponent, you can apply the exponent to each factor in the product individually. Mathematically, this is expressed as: (a×b)n=an×bn (a \times b)^n = a^n \times b^n .

In this case, the expression (2×3)2 \left(2\times3\right)^2 follows this rule with a=2 a = 2 and b=3 b = 3 , and n=2 n = 2 .

  • First, apply the exponent to the first factor: 22 2^2 .
  • Next, apply the exponent to the second factor: 32 3^2 .

Therefore, by applying the "Power of a Product" rule, the expression becomes: 22×32 2^2 \times 3^2 .

Answer

22×32 2^2\times3^2

Exercise #10

Insert the corresponding expression:

(2×4)10= \left(2\times4\right)^{10}=

Video Solution

Step-by-Step Solution

To solve the question, we apply the rule of exponents known as the Power of a Product. The formula states that for any real numbers a a and b b , and an integer n n :

  • (a×b)n=an×bn (a \times b)^n = a^n \times b^n

Given the expression (2×4)10 (2 \times 4)^{10} , we can identify:

  • a=2 a = 2
  • b=4 b = 4
  • n=10 n = 10

Now, applying the formula:

  • (2×4)10=210×410 (2 \times 4)^{10} = 2^{10} \times 4^{10}

Thus, the expression (2×4)10 (2 \times 4)^{10} is equivalent to 210×410 2^{10} \times 4^{10} .

Answer

210×410 2^{10}\times4^{10}

Exercise #11

Insert the corresponding expression:

(5×6)9= \left(5\times6\right)^9=

Video Solution

Step-by-Step Solution

To solve the expression (5×6)9 \left(5\times6\right)^9 , we apply the rule for the power of a product. This rule states that when you raise a product to a power, it is equivalent to raising each factor in the product to the same power. Mathematically, this is expressed as:

  • (a×b)n=an×bn (a \times b)^n = a^n \times b^n

In this specific problem, the factors are 5 5 and 6 6 and the exponent is 9 9 . Applying the power of a product rule, we get:

  • (5×6)9=59×69 \left(5\times6\right)^9 = 5^9 \times 6^9

Therefore, the correct expression that corresponds to (5×6)9 \left(5\times6\right)^9 is 59×69 5^9\times6^9 .

Answer

59×69 5^9\times6^9

Exercise #12

Insert the corresponding expression:

(13×4)6= \left(13\times4\right)^6=

Video Solution

Step-by-Step Solution

The given expression to solve is (13×4)6 \left(13\times4\right)^6 . This expression involves raising a product to a power. According to the exponent rule, which is the Power of a Product Rule, the property can be stated as follows:


  • When you have a product raised to an exponent, you can distribute the exponent to each factor in the product separately.

Mathematically, for any numbers aa and bb, and a positive integer nn, it is written as: (a×b)n=an×bn (a \times b)^n = a^n \times b^n .


Applying this rule to our expression:


(13×4)6 \left(13 \times 4\right)^6 becomes 136×46 13^6 \times 4^6 .


Thus, the expression (13×4)6 \left(13 \times 4\right)^6 simplifies to 136×46 13^6 \times 4^6 following the Power of a Product Rule.

Answer

136×46 13^6\times4^6

Exercise #13

Insert the corresponding expression:

(12×3)5= \left(12\times3\right)^5=

Video Solution

Step-by-Step Solution

The given expression is (12×3)5 (12 \times 3)^5 . According to the Power of a Product rule, which states that (a×b)n=an×bn(a \times b)^n = a^n \times b^n, we apply this formula to the expression.


  • Firstly, identify the base of the power as the product 12×312 \times 3.
  • Secondly, recognize that the exponent applied to this product is 5.
  • According to the rule, the power of a product can be distributed to each factor in the product, which means: (12×3)5=125×35(12 \times 3)^5 = 12^5 \times 3^5.

Therefore, the expression (12×3)5 (12 \times 3)^5 corresponds to 125×35 12^5 \times 3^5 .

Answer

125×35 12^5\times3^5

Exercise #14

(2×8×7)2= (2\times8\times7)^2=

Video Solution

Step-by-Step Solution

We begin by using the power rule for parentheses:

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n

That is, the power applied to a product inside parentheses, is applied to each of the terms within, when the parentheses are opened.

We then apply the above rule to the problem:

(287)2=228272 (2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2

Therefore, the correct answer is option d.

Note:

From the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms inside parentheses (as formulated above), then it is also valid for a power over several terms of the product inside parentheses (for example - three terms, etc.).

Answer

228272 2^2\cdot8^2\cdot7^2

Exercise #15

(2×7×5)3= (2\times7\times5)^3=

Video Solution

Step-by-Step Solution

To solve the problem(2×7×5)3 (2\times7\times5)^3 , we need to apply the Power of a Product rule of exponents. This rule states that when you raise a product to a power, it's the same as raising each factor to that power. In mathematical terms, if you have (abc)n (abc)^n , it is equivalent to an×bn×cn a^n \times b^n \times c^n .

Let's apply this rule step by step:

Our original expression is (2×7×5)3 (2 \times 7 \times 5)^3 .

We identify the factors inside the parentheses as 2 2 , 7 7 , and 5 5 .

According to the Power of a Product rule, we can distribute the exponent3 3 to each factor:

First, raise 2 2 to the power of 3 3 to get 23 2^3 .

Then, raise 7 7 to the power of 3 3 to get 73 7^3 .

Finally, raise 5 5 to the power of 3 3 to get 53 5^3 .

Therefore, the expression (2×7×5)3 (2 \times 7 \times 5)^3 simplifies to 23×73×53 2^3 \times 7^3 \times 5^3 .

Answer

23×73×53 2^3\times7^3\times5^3