To simplify this expression, we will need to multiply the exponents of the inside quantity by the outer exponent.

\(\displaystyle (3x^4)^{3} = (3^{1}\cdot x^4)^{3} = (3^{1\cdot 3 }\cdot x^{4\cdot 3})\)

Simplify the terms by multiplication.

\(\displaystyle 3^3 \cdot x^{12} = 27x^{12}\)

The answer is:  \(\displaystyle 27x^{12}\)

In order to solve this expression, use the power rule of exponents.

\(\displaystyle (x^a)^b = x^{ab}\)

Notice that there is also a two in front of the variable.  We will need to convert that as an exponent as well.

\(\displaystyle (2a^8)^{3}=(2^1\cdot a^8)^{3}= 2^3\cdot a^{8\times 3}\)

Simplify the expression.

The answer is:  \(\displaystyle 8a^{24}\)

To evaluate this expression, we will need to convert to common bases in order to use the power rule for exponents.

\(\displaystyle 4=2^2\)

\(\displaystyle 8=2^3\)

Rewrite the problem.

\(\displaystyle 2(4^{10})(8^{20})^3 =2(2^{2(10)})(2^{3(20)})^3= 2\cdot 2^{20}\cdot(2^{60})^3\)

Do not mix the additive rule of exponents with the multiplicative rule.

\(\displaystyle a^b\cdot a^c = a^{b+c}\)

\(\displaystyle (x^y)^z = x^{yz}\)

Use these rules to simplify the terms.

\(\displaystyle 2\cdot 2^{20}\cdot(2^{60})^3 = 2^1\cdot 2^{20}\cdot2^{180} = 2^{1+20+180} =2^{201}\)

The answer is:  \(\displaystyle 2^{201}\)

Use the power rule of exponents to simplify this expression.

\(\displaystyle (x^{Y})^Z = x^{YZ}\)

Follow this rule to simplify the exponents.

\(\displaystyle (8^{4})^{15} = 8^{4\times 15} = 8^{60}\)

The answer is:  \(\displaystyle 8^{60}\)

We can use the power rule of exponents to simplify this expression.

\(\displaystyle (x^{Y})^Z =x^{YZ}\)

\(\displaystyle [(2^{-3})^{-2}]^{-\frac{1}{2}} = 2^{(-3)\cdot (-2 )\cdot( -\frac{1}{2})}\)

The expression becomes:

\(\displaystyle 2^{-3}\)

We can rewrite this as a fraction.

\(\displaystyle 2^{-3} = \frac{1}{2^3}=\frac{1}{8}\)

The answer is:  \(\displaystyle \frac{1}{8}\)

Use the product rule of exponents to simplify the terms in parentheses.

\(\displaystyle (b^{X})^Y= b^{XY}\)

\(\displaystyle b(b^{-3})^{-10} = b(b^{-3\times -10}) = b(b^{30})\)

This expression is the same as:  \(\displaystyle b^1 \cdot b^{30}\)

The answer is:  \(\displaystyle b^{31}\)

In order to solve this expression, we will need to use the power rule of exponents.

\(\displaystyle (a^{x})^y = a^{xy}\)

We can distribute the outer exponent to the exponents inside the parentheses.  Do not ignore the five, which equals \(\displaystyle 5^1\).

\(\displaystyle (5x^3)^4 = (5^{1\times 4}\cdot x^{3\times 4}) = 5^4 \cdot x^{12}\)

The answer is:  \(\displaystyle 625x^{12}\)

In order to solve this, we can change this expression to base two.

The fraction can be rewritten as a negative exponent.

\(\displaystyle \frac{1}{2} =2^{-1}\)

The inner term in the parentheses is equal to:  

\(\displaystyle 64= 2^6\)

Rewrite the expression.

\(\displaystyle \frac{1}{2}(64^{5})^{10} = 2^{-1} (2^{6(5)})^{(10)}\)

Using the product rule of exponents, we can multiply all the powers together.

\(\displaystyle 2^{-1} (2^{6\times 5\times 10})= 2^{-1} (2^{300}) = 2^{-1} \cdot (2^{300})\)

Now that we are multiplying two common bases with exponents, we can add the powers.

\(\displaystyle 2^{-1+300} = 2^{299}\)

The answer is:  \(\displaystyle 2^{299}\)

The exponents that are surrounded by the brackets and parentheses can be multiplied based on the product rule for exponents.

\(\displaystyle 5[(5^3)^4]^5 = 5^1\cdot [5^{3\times 4\times 5}] = 5^1\cdot 5^{60}\)

Now that we have bases similar, we can use the additive rule of exponents to combine both terms.

The answer is:  \(\displaystyle 5^{61}\)

According to the product rule of exponents, we can multiply the two powers together in the parentheses.

\(\displaystyle (x^B)^C = x^{B\times C}\)

This means that:

\(\displaystyle (3^{-3})^2 = 3^{-6}\)

Replace this term in the bracket.

\(\displaystyle [(3^{-3})^2]^{\frac{1}{2}} = [3^{-6}]^{\frac{1}{2}}\)

Repeat the process for these exponents.

\(\displaystyle [3^{-6}]^{\frac{1}{2}} = 3^{-6\times \frac{1}{2}} = 3^{-3}\)

This negative exponent can be simplified as a fraction.

\(\displaystyle 3^{-3} = \frac{1}{3^3} = \frac{1}{27}\)

The answer is:  \(\displaystyle \frac{1}{27}\)