Multiplication of Powers - Examples, Exercises and Solutions

$4^2\times4^4=$

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

$a^n * a^m = a^{n+m}$

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

$4^2\times4^4=4^{4+2}=4^6$

$4^6$

$7^9\times7=$

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:$a^n\times a^m=a^{n+m}$

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:

$7^{9+1}=7^{10}$

$7^{10}$

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

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

$8^2\cdot8^3\cdot8^5=8^{10}$

$8^{10}$

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

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

$a^m\cdot a^n=a^{m+n}$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:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$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:

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

$2^{23}$

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

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

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

$3^{3x}\cdot2^x$

$a^3\times a^4=$

Here, we will need to calculate a multiplication between terms with identical bases, therefore we will use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$Note that this property can only be used to calculate the multiplication between terms with identical bases,

We apply it to the problem:

$a^3\cdot a^4=a^{3+4}=a^7$Therefore, the correct answer is option b.

$a^7$

$x^2\times x^5=$

Here we will have to to multiply terms with identical bases, therefore we use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$Note that this property can only be used to calculate the multiplication between terms with identical bases,

From now on we no longer write the multiplication sign, but use the accepted form of writing in which placing terms next to each other means multiplication.

We apply it in the problem:

$x^2x^5=x^{2+5}=x^7$Therefore, the correct answer is D.

$x^7$

$5^4\times25=$

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.

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

$a^n\times a^m=a^{n+m}$

$5^4\times25=5^4\times5^2=5^{4+2}=5^6$

$5^6$

$2^{2x+1}\cdot2^5\cdot2^{3x}=$

Since the bases are the same, the exponents can be added:

$2x+1+5+3x=5x+6$

$2^{5x+6}$

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6=$

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

$a^m\cdot a^n=a^{m+n}$We apply the property for this problem:

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6= 4^{2y+(-5)+(-y)+6}=4^{2y-5-y+6}$We simplify the expression we got in the last step:

$4^{2y-5-y+6} =4^{y+1}$When we add similar terms in the exponent.

Therefore, the correct answer is option c.

$4^{y+1}$

$7^{2x+1}\cdot7^{-1}\cdot7^x=$

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

$a^m\cdot a^n=a^{m+n}$We apply the property to our expression:

$7^{2x+1}\cdot7^{-1}\cdot7^x=7^{2x+1+(-1)+x}=7^{2x+1-1+x}$We simplify the expression we got in the last step:

$7^{2x+1-1+x}=7^{3x}$When we add similar terms in the exponent.

Therefore, the correct answer is option d.

$7^{3x}$

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

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

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

$7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1}$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:

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

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

$$$\frac{1}{7}$

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

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

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

$12^4\cdot12^{-6}=12^{4+(-6)}=12^{4-6}=12^{-2}$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:

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

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

$\frac{1}{144}$

$7^x\cdot7^{-x}=\text{?}$

We use the law of exponents to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the law to given the problem:

$7^x\cdot7^{-x}=7^{x+(-x)}=7^{x-x}=7^0$In the first stage we apply the above power rule and in the following stages we simplify the expression obtained in the exponent,

Subsequently, we use the zero power rule:

$X^0=1$We obtain:

$7^0=1$Lastly we summarize the solution to the problem.

$7^x\cdot7^{-x}=7^{x-x}=7^0 =1$Therefore, the correct answer is option B.

$1$

Solve the exercise:

$Y^2+Y^6-Y^5\cdot Y=$

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

$a^m\cdot a^n=a^{m+n}$We apply it in the problem:

$Y^2+Y^6-Y^5\cdot Y=Y^2+Y^6-Y^{5+1}=Y^2+Y^6-Y^6=Y^2$When we apply the previous property to the third expression from the left in the sum, and then simplify the total expression by adding like terms.

Therefore, the correct answer is option D.

$Y^2$