# Power of a power - Examples, Exercises and Solutions

## Power of a Power

When we have an expression raised to a power that, in turn, is raised (within parentheses) to another power, we can multiply the exponents and raise the base number to the result of this multiplication.

### Formula of the property

$(a^n)^m=a^{(n\times m)}$
This property is also concerning algebraic expressions.

## Practice Power of a power

### Exercise #1

Solve the exercise:

$(a^5)^7=$

### Step-by-Step Solution

We use the formula:

$(a^m)^n=a^{m\times n}$

and therefore we obtain:

$(a^5)^7=a^{5\times7}=a^{35}$

$a^{35}$

### Exercise #2

$(4^2)^3+(g^3)^4=$

### Step-by-Step Solution

We use the formula:

$(a^m)^n=a^{m\times n}$

$(4^2)^3+(g^3)^4=4^{2\times3}+g^{3\times4}=4^6+g^{12}$

$4^6+g^{12}$

### Exercise #3

$(a^4)^6=$

### Step-by-Step Solution

We use the formula

$(a^m)^n=a^{m\times n}$

Therefore, we obtain:

$a^{4\times6}=a^{24}$

$a^{24}$

### Exercise #4

$[(\frac{1}{7})^{-1}]^4=$

### Step-by-Step Solution

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$Therefore, the correct answer is option c

$7^4$

### Exercise #5

$((y^6)^8)^9=$

### Step-by-Step Solution

We use the property of powers of an exponent raised to another exponent:

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

$\big((y^6)^8\big)^9=(y^{6\cdot8})^9=y^{6\cdot8\cdot9}=y^{432}$When we use the aforementioned property twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the last stage we calculate the result of the multiplication in the power exponent.

Therefore, the correct answer is option b.

$y^{432}$

### Exercise #1

$(4^x)^y=$

### Step-by-Step Solution

Using the law of powers for an exponent raised to another exponent:

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

$(4^x)^y=4^{xy}$Therefore, the correct answer is option a.

$4^{xy}$

### Exercise #2

Solve the exercise:

$(x^2\times3)^2=$

### Step-by-Step Solution

We have an exponent raised to another exponent with a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$This says that in a case where a power is applied to a multiplication between parentheses,the power is applied to each term of the multiplication when the parentheses are opened,

We apply it in the problem:

$(3x^2)^2=3^2(x^2)^2$With the second term of the multiplication we proceed carefully, since it is already in a power (that's why we use parentheses). The term will be raised using the power law for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$and we apply it in the problem:

$3^2(x^2)^2=9x^{2\cdot2}=9x^4$In the first step we raise the number to the power, and in the second step we multiply the exponent.

Therefore, the correct answer is option a.

$9x^4$

### Exercise #3

$(2^2)^3+(3^3)^4+(9^2)^6=$

### Step-by-Step Solution

We use the formula:

$(a^m)^n=a^{m\times n}$

$(2^2)^3+(3^3)^4+(9^2)^6=2^{2\times3}+3^{3\times4}+9^{2\times6}=2^6+3^{12}+9^{12}$

$2^6+3^{12}+9^{12}$

### Exercise #4

$((b^3)^6)^2=$

### Step-by-Step Solution

We use the formula

$(a^m)^n=a^{m\times n}$

Therefore, we obtain:

$((b^3)^6)^2=(b^{3\times6})^2=(b^{18})^2=b^{18\times2}=b^{36}$

$b^{36}$

### Exercise #5

$((a^2)^3)^{\frac{1}{4}}=$

### Step-by-Step Solution

We use the property of powers of an exponent raised to another exponent:

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

$\big((a^2)^3\big)^{\frac{1}{4}}=(a^{2\cdot3})^{\frac{1}{4}}=a^{2\cdot3\cdot\frac{1}{4}}=a^{\frac{6}{4}}=a^{\frac{3}{2}}$When we use the previously mentioned property twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the third stage we calculate the result of the multiplication in the exponent. While remembering that multiplying by a fraction is actually doubling the numerator of the fraction and, finally, in the last stage we simplify the fraction we obtained in the exponent.

Now remember that -

$\frac{3}{2}=1\frac{1}{2}=1.5$

Therefore, the correct answer is option a.

$a^{1.5}$

### Exercise #1

$((14^{3x})^{2y})^{5a}=$

### Step-by-Step Solution

Using the property of powers of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply the law in the expression of the problem:

$((14^{3x})^{2y})^{5a}=(14^{3x})^{2y\cdot5a}=14^{3x\cdot2y\cdot5a}=14^{30xya}$When in the first step we applied the aforementioned power property and got rid of the outer parentheses, in the next step we applied the power property in question again and got rid of the remaining parentheses, in the next step we simplified the resulting expression,

Therefore, from the use of the substitution property in multiplication (which is applied to the exponent of the power in the obtained expression) it can be concluded that the correct answer is answer D.

$14^{30axy}$

### Exercise #2

$((3^9)^{4x)^{5y}}=$

### Step-by-Step Solution

We use the power property for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply this law to the expression of the problem:

$((3^9)^{4x})^{5y}= (3^9)^{4x\cdot 5y} =3^{9\cdot4x\cdot 5y}=3^{180xy}$When in the first step we applied the previously mentioned power property and got rid of the outer parentheses, in the next step we applied the power property in question again and got rid of the remaining parentheses, in the next step we simplified the resulting expression.

Therefore, the correct answer is option b.

$3^{180xy}$

### Exercise #3

$(y^3\times x^2)^4=$

### Step-by-Step Solution

We will solve it in two steps, in the first step we will use the power of a product within parentheses law:

$(z\cdot t)^n=z^n\cdot t^n$The one that states that the power affecting a product within parentheses applies to each of the elements of the product when the parentheses are opened,

We apply the law to the problem:

$(y^3\cdot x^2)^4=(y^3)^4\cdot(x^2)^4$When we open the parentheses, we apply the power to each of the terms of the product separately, but since each of these terms is already raised to a power, we did it cautiously and used parentheses.

Then, we will use the law of raising a power to another power

$(b^m)^n=b^{m\cdot n}$We apply the law to the problem we obtained:

$(y^3)^4\cdot(x^2)^4=y^{3\cdot4}\cdot x^{2\cdot4}=y^{12}\cdot x^8$When in the second step we perform the multiplication operation on the power exponents of the obtained terms.

Therefore, the correct answer is option d.

$y^{12}x^8$

### Exercise #4

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

### Step-by-Step Solution

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

First we use the law

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

After that, we will open the parentheses according to the law.

$\left(abc\right)^x=a^x\cdot b^x\cdot c^x$

$8^{3y}\cdot b^{3y}\cdot y^{3y}+3^{xa}$

$8^{3y}\times b^{3y}\times y^{3y}+3^{ax}$

### Exercise #5

$(4\cdot7)^9+\frac{2^7}{2^4}+(8^2)^5=$

### Step-by-Step Solution

To solve the problem we use two power laws, remember them:

A. Power property for terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$B. Power property for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We will apply these two power laws to the problem expression in two steps:

Let's start and apply the power law specified in A to the second term from the left in the problem expression:

$\frac{2^7}{2^4}=2^{7-4}=2^3$When in the first step we apply the power law specified in A and in the following steps we simplify the resulting expression,

We will proceed to the next step and apply the power law specified in B and address the third term from the left in the problem expression:

$(8^2)^5=8^{2\cdot5}=8^{10}$When in the first stage we apply the power law specified in B and in the following stages we simplify the resulting expression,

Let's summarize the two steps listed above to solve the general problem:

$(4\cdot7)^9+\frac{2^7}{2^4}+(8^2)^5= (4\cdot7)^9+2^3+8^{10}$In the next step, we calculate the result of multiplying the terms within the parentheses in the first term from the left:

$(4\cdot7)^9+2^3+8^{10}=28^9+2^3+8^{10}$Therefore, the correct answer is option c.

$28^9+2^3+8^{10}$