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:

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

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

$5^6$

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

First we use the power property for a multiplication between terms with identical bases:

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

$7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1}$When in a first stage we apply the aforementioned property and then simplify the expression in the exponent,

Next, we use the property of negative powers:

$a^{-n}=\frac{1}{a^n}$And we apply it in the expression we obtained in the last step:

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

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

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

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.

$7^{3x}$

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

First we use the power property for a multiplication between terms with identical bases:

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

$12^4\cdot12^{-6}=12^{4+(-6)}=12^{4-6}=12^{-2}$When in a first stage we apply the aforementioned property and then simplify the expression in the exponent,

Next, we use the property of negative powers:

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

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

$\frac{1}{144}$

$5^4\cdot(\frac{1}{5})^4=\text{?}$

This problem can be solved using the properties of powers for a negative power, power over a power, and the property of powers for the product between terms with identical bases, which is the natural way of solving,

But here we prefer to solve it in another way that is a bit faster:

To this end, the power by power law is applied to the parentheses in which the terms are multiplied, but in the opposite direction:

$x^n\cdot y^n=(x\cdot y)^n$Since in the expression in the problem there is a multiplication between two terms with identical powers, this law can be used in its opposite sense, so we will apply this property to the problem:

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4$Since the multiplication in the given problem is between terms with the same power, we could apply this law in the opposite direction and write the expression as the multiplication of the bases of the terms in parentheses to which the same power is applied.

We will continue and simplify the expression in parentheses, we will do it quickly if we notice that in parentheses there is a multiplication between two opposite numbers, then their product will give the result: 1, we will apply this understanding to the expression we arrived at in the last step:

$\big(5\cdot\frac{1}{5}\big)^4 = 1^4=1$When in the first step we apply the previous understanding, and then use the fact that raising the number 1 to any power will always give the result: 1, which means that:

$1^x=1$Summarizing the steps to solve the problem, we get that:

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4 =1$Therefore, the correct answer is option b.

1

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

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

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

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

Subsequently, we use the zero power property:

$X^0=1$We obtain:

$7^0=1$We summarize the solution to the problem, we obtained that:

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

$1$