Multiplication of powers - Examples, Exercises and Solutions

When we are presented with exercises or expressions where multiplication of powers with the same base appears, we can add the exponents.

The result obtained from adding the exponents will be the new exponent and the original base is maintained.

The formula of the rule:
am×an=a(m+n) a^m\times a^n=a^{(m+n)}

It doesn't matter how many terms there are. As long as there are products of powers with the same base, we can add their exponents and obtain a new one that we apply to the base.

It is important to remember that this property should only be applied when there are products of powers with the same base. In other words, if we have a multiplication of powers with different bases, we cannot add the exponents.

This property also pertains to algebraic expressions.

Suggested Topics to Practice in Advance

  1. Power of a Power
  2. Exponent of a Multiplication
  3. Power of a Quotient

Practice Multiplication of powers

Exercise #1

42×44= 4^2\times4^4=

Video Solution

Step-by-Step Solution

To solve the exercise we use the property of multiplication of powers with the same bases:

anam=an+m a^n * a^m = a^{n+m}

With the help of this property, we can add the exponents.

42×44=44+2=46 4^2\times4^4=4^{4+2}=4^6

Answer

46 4^6

Exercise #2

79×7= 7^9\times7=

Video Solution

Step-by-Step Solution

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:an×am=an+m a^n\times a^m=a^{n+m}

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

79+1=710 7^{9+1}=7^{10}

Answer

710 7^{10}

Exercise #3

(35)4= (3^5)^4=

Video Solution

Step-by-Step Solution

To solve the exercise we use the power property:(an)m=anm (a^n)^m=a^{n\cdot m}

We use the property with our exercise and solve:

(35)4=35×4=320 (3^5)^4=3^{5\times4}=3^{20}

Answer

320 3^{20}

Exercise #4

(62)13= (6^2)^{13}=

Video Solution

Step-by-Step Solution

We use the formula:

(an)m=an×m (a^n)^m=a^{n\times m}

Therefore, we obtain:

62×13=626 6^{2\times13}=6^{26}

Answer

626 6^{26}

Exercise #5

828385= 8^2\cdot8^3\cdot8^5=

Video Solution

Step-by-Step Solution

All bases are equal and therefore the exponents can be added together.

828385=810 8^2\cdot8^3\cdot8^5=8^{10}

Answer

810 8^{10}

Exercise #1

2102726= 2^{10}\cdot2^7\cdot2^6=

Video Solution

Step-by-Step Solution

We use the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

amanak=am+nak=am+n+k a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k} When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms in the multiplication have the same base, so we will use the previous property:

2102726=210+7+6=223 2^{10}\cdot2^7\cdot2^6=2^{10+7+6}=2^{23} Therefore, the correct answer is option c.

Answer

223 2^{23}

Exercise #2

54×25= 5^4\times25=

Video Solution

Step-by-Step Solution

To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.

25=5 \sqrt{25}=5 25=52 25=5^2 Now, we go back to the initial exercise and solve by adding the powers according to the formula:

an×am=an+m a^n\times a^m=a^{n+m}

54×25=54×52=54+2=56 5^4\times25=5^4\times5^2=5^{4+2}=5^6

Answer

56 5^6

Exercise #3

Simplify the expression:

a3a2b4b5= a^3\cdot a^2\cdot b^4\cdot b^5=

Video Solution

Step-by-Step Solution

In the exercise of multiplying powers, we will add up all the powers of the same product, in this case the terms a, b

We use the formula:

an×am=an+m a^n\times a^m=a^{n+m}

We are going to focus on the term a:

a3×a2=a3+2=a5 a^3\times a^2=a^{3+2}=a^5

We are going to focus on the term b:

b4×b5=b4+5=b9 b^4\times b^5=b^{4+5}=b^9

Therefore, the exercise that will be obtained after simplification is:

a5×b9 a^5\times b^9

Answer

a5b9 a^5\cdot b^9

Exercise #4

k2t4k6t2= k^2\cdot t^4\cdot k^6\cdot t^2=

Video Solution

Step-by-Step Solution

Using the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} It is important to note that this law is only valid for terms with identical bases,

We notice that in the problem there are two types of terms. First, for the sake of order, we will use the substitution property to rearrange the expression so that the two terms with the same base are grouped together. The, we will proceed to solve:

k2t4k6t2=k2k6t4t2 k^2t^4k^6t^2=k^2k^6t^4t^2 Next, we apply the power property to each different type of term separately,

k2k6t4t2=k2+6t4+2=k8t6 k^2k^6t^4t^2=k^{2+6}t^{4+2}=k^8t^6 We apply the property separately - for the terms whose bases arek k and for the terms whose bases aret t We add the powers in the exponent when we multiply all the terms with the same base.

The correct answer then is option b.

Answer

k8t6 k^8\cdot t^6

Exercise #5

ababa2 a\cdot b\cdot a\cdot b\cdot a^2

Video Solution

Step-by-Step Solution

We use the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} It is important to note that this property is only valid for terms with identical bases,

We return to the problem

We notice that in the problem there are two types of terms with different bases. First, for the sake of order, we will use the substitution property of multiplication to rearrange the expression so that the two terms with the same base are grouped together. Then, we will proceed to work:

ababa2=aaa2bb a\cdot b\operatorname{\cdot}a\operatorname{\cdot}b\operatorname{\cdot}a^2=a\cdot a\cdot a^2\cdot b\cdot b Next, we apply the power property for each type of term separately,

aaa2bb=a1+1+2b1+1=a4b2 a\cdot a\cdot a^2\cdot b\cdot b=a^{1+1+2}\cdot b^{1+1}=a^4\cdot b^2

We apply the power property separately - for the terms whose bases area a and then for the terms whose bases areb b and we add the exponents and simplify the terms.

Therefore, the correct answer is option c.

Note:

We use the fact that:

a=a1 a=a^1 and the same for b b .

Answer

a4b2 a^4\cdot b^2

Exercise #1

E6F4E0F7E= E^6\cdot F^{-4}\cdot E^0\cdot F^7\cdot E=

Video Solution

Step-by-Step Solution

We use the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} It should be noted that this property is only valid for terms with identical bases,

We return to the problem

We notice that in the problem there are two types of terms with different bases. First, for the sake of order, we will use the substitution property of multiplication to rearrange the expression so that the two terms with the same base are grouped together. Then, we will proceed to work:

E6F4E0F7E=E6E0EF4F7 E^6\cdot F^{-4}\cdot E^0\cdot F^7\cdot E=E^6\cdot E^0\cdot E\cdot F^{-4}\cdot F^7 Next, we apply the power property for each type of term separately,

E6E0EF4F7=E6+0+1F4+7=E7F3 E^6\cdot E^0\cdot E\cdot F^{-4}\cdot F^7=E^{6+0+1}\cdot F^{-4+7}=E^7\cdot F^3

We apply the power property separately - for the terms whose bases areE E and for the terms whose bases areF F and we add the exponents and simplify the terms with the same base.

The correct answer is then option d.

Note:

We use the fact that:

E=E1 E=E^1 .

Answer

E7F3 E^7\cdot F^3

Exercise #2

9380=? \frac{9\cdot3}{8^0}=\text{?}

Video Solution

Step-by-Step Solution

We use the formula:

a0=1 a^0=1

9×380=9×31=9×3 \frac{9\times3}{8^0}=\frac{9\times3}{1}=9\times3

We know that:

9=32 9=3^2

Therefore, we obtain:

32×3=32×31 3^2\times3=3^2\times3^1

We use the formula:

am×an=am+n a^m\times a^n=a^{m+n}

32×31=32+1=33 3^2\times3^1=3^{2+1}=3^3

Answer

33 3^3

Exercise #3

53505255= 5^{-3}\cdot5^0\cdot5^2\cdot5^5=

Video Solution

Step-by-Step Solution

We use the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

amanak=am+nak=am+n+k a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k} When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms of the multiplication have the same base, so we will use the previous property:

53505255=53+0+2+5=54 5^{-3}\cdot5^0\cdot5^2\cdot5^5=5^{-3+0+2+5}=5^4 Therefore, the correct answer is option c.

Note:

Keep in mind that 50=1 5^0=1

Answer

54 5^4

Exercise #4

101021041010= 10\cdot10^2\cdot10^{-4}\cdot10^{10}=

Video Solution

Step-by-Step Solution

We use the power property to multiply terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n} Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

amanak=am+nak=am+n+k a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k} When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

First keep in mind that:

10=101 10=10^1 Keep in mind that all the terms of the multiplication have the same base, so we will use the previous property:

1011021041010=101+24+10=109 10^1\cdot10^2\cdot10^{-4}\cdot10^{10}=10^{1+2-4+10}=10^9

Therefore, the correct answer is option c.

Answer

109 10^9

Exercise #5

3x2x32x= 3^x\cdot2^x\cdot3^{2x}=

Video Solution

Step-by-Step Solution

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of 3 3

3x2x32x=2x33x 3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}

Answer

33x2x 3^{3x}\cdot2^x

Topics learned in later sections

  1. Division of Exponents with the Same Base
  2. Rules of Exponentiation
  3. Combining Powers and Roots