When an exponent is being raised by another exponent, we multiply the exponents and keep the base the same. Since there is a number present, we also must perform the exponent operation as well.

\(\displaystyle (3x^6)^5=3^5*x^{5*6}=243x^{30}\)

Since we have two expressions being multiplied, we follow PEMDAS meaning we do exponents first then multiplication. We will apply power rule and then multiplication rule of exponents.

\(\displaystyle 4x^2(3x^6)^2=4x^2*9x^{12}=36x^{14}\)

Since we have two expressions being multiplied, we follow PEMDAS meaning we do exponents first then multiplication. We will apply power rule and then multiplication rule of exponents.

\(\displaystyle 5x^6(4x^2)^3=5x^6*64x^6=320x^{12}\)

Evaluate by using the product rule of exponents.  Solve the parentheses first.

\(\displaystyle (9^{4})^{20} = 9^{4\cdot 20} =9^{80}\)

Solve the bracket.

\(\displaystyle [(9^{4})^{20}]^2=[9^{80}]^2 = 9^{80\cdot 2} =9^{160}\)

The answer is:  \(\displaystyle 9^{160}\)

Recall that when an exponent is raised to another exponent, you will need to multiply the two exponents together.

Start by simplifying the numerator:

\(\displaystyle (x^2y^3)^9=x^{18}y^{27}\)

Now, place this on top of the denominator and simplify. Recall that when you divide exponents that have the same base, you will subtract the exponent in the denominator from the exponent in the numerator.

\(\displaystyle \frac{x^{18}y^{27}}{xy^7}=x^{17}y^{20}\)

We can solve the exponent in the parentheses first.

Use the power rule to simplify the term.

\(\displaystyle (2^{20})^5 = (2^{20\cdot 5}) = 2^{100}\)

The expression becomes:  

\(\displaystyle 2(2^{20})^5 = 2(2^{100}) = 2^1 \cdot 2^{100}\)

Since the bases of the powers are the same, we can add the exponents.

The answer is:  \(\displaystyle 2^{101}\)

Simplify the powers in parentheses first.  According to the product rule, their powers can be multiplied.

\(\displaystyle (25^2)^3 = 25^6\)

The expression becomes:

\(\displaystyle 5^4(25^2)^3=5^4(25^6)\)

Convert the number 25 to base five.

\(\displaystyle 5^4(5^{2(6)}) = 5^4(5^{12})\)

Since the bases are similar, the property of exponents allow us to add the exponents.

The answer is:  \(\displaystyle 5^{16}\)

The exponents are separated by a parentheses.

According to the distributive property of exponents, we can simply multiply the two exponents.

\(\displaystyle 3^{600\times 3} = 3^{1800}\)

Do not cube the three!

The answer is:  \(\displaystyle 3^{1800}\)

To simplify this expression, convert the value of 36 to base six.

\(\displaystyle 6(36^{10})^3 = 6(6^{2(10)})^3 = 6(6^{20})^3\)

Evaluate the product of exponents.

\(\displaystyle 6(6^{60})\)

This is similar to: \(\displaystyle 6^1 \cdot 6^{60}\)

When common bases of a certain powers are multiplied, the exponents can be added.

The answer is:  \(\displaystyle 6^{61}\)

Evaluate the expression by using the product rule of exponents.

\(\displaystyle 2(2^{19})^4 =2(2^{19\times 4})= 2(2^{76})\)

Rewrite the expression using \(\displaystyle 2=2^1\).

\(\displaystyle 2(2^{76}) = 2^1 \cdot 2^{76}\)

When powers of a same base are multiplied together, we can add the exponents.

The answer is:  \(\displaystyle 2^{77}\)