To solve this expression, we can use the power property of exponents to simplify the powers.

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

Use order of operations to solve the inner term.  Since negative three in not in a quantity itself, only the number three is raised to the power of six.

\(\displaystyle -(-3^6) = -(-729)= 729\)

The answer is:  \(\displaystyle 729\)

Evaluate the inner parentheses first.  We can use the product rule of exponents to simplify this term.

\(\displaystyle (2^{30})^5 = 2^{30\times 5} = 2^{150}\)

Replace the term in the bracket.

\(\displaystyle [4(2^{30})^5]^3= [4(2^{150})]^3\)

Convert the base of the value of 4 to base two.

\(\displaystyle 4=2^2\)

\(\displaystyle [4(2^{150})]^3 = [2^2(2^{150})]^3 = [2^2\cdot2^{150}]^3\)

The inner bracket shares common bases of a power that are multiplied.  The rule of exponents allow us to add the terms.

\(\displaystyle [2^2\cdot2^{150}]^3 = [2^{2+150}]^3 = [2^{152}]^3\)

Use the product rule to simplify this term.

\(\displaystyle [2^{152}]^3= 2^{152\times 3}\)

The answer is:  \(\displaystyle 2^{456}\)

In order to determine the value of this expression, change all the bases to base two.

\(\displaystyle 8(2^{30})^3(4^{10})^3=(2^3)(2^{30})^3(2^{2(10)})^3 =(2^3)(2^{30})^3(2^{20})^3\)

Simplify the exponents separated by the inside and outside of the ending parentheses by multiplying the terms.

\(\displaystyle (2^3)(2^{30})^3(2^{20})^3 = (2^3)\cdot (2^{30\times 3})\cdot (2^{20\times 3})\)

Multiply the exponents.

\(\displaystyle (2^3)\cdot (2^{30\times 3})\cdot (2^{20\times 3})= (2^3)\cdot (2^{90})\cdot (2^{60})\)

Now that we are multiplying the similar bases with exponents, the exponents can be added.

\(\displaystyle (2^3)\cdot (2^{90})\cdot (2^{60})=2^{3+90+60} = 2^{153}\)

The answer is:  \(\displaystyle 2^{153}\)

In order to evaluate this expression, we can use the distributive property of exponents to simplify.

Multiply the powers together.

\(\displaystyle 2(2^2)^4 = 2(2)^{2(4)} = 2(2^8)\)

This is also the same as:  \(\displaystyle 2^1 \cdot 2^8 = 2^{1+8} = 2^9\)

\(\displaystyle 2^9 = (2\cdot2\cdot2)^3 = 8^3 =512\)

The answer is: \(\displaystyle 512\)

Solve by multiplying the exponents of each term inside the parentheses by the outer exponent of the parentheses.

The exponent of three inside the parentheses can be written as a one.

\(\displaystyle (3x^{45})^3=(3^1x^{45})^3 = 3^3 x^{45\times 3} = 27x^{135}\)

The answer is:  \(\displaystyle 27x^{135}\)

We will need to use the power rule of exponents to simplify the term enclosed in the parentheses.

\(\displaystyle (a^b)^c = a^{bc}\)

\(\displaystyle b(b^{15})^3 = b\cdot b^{15\times3} = b\cdot b^{45}\)

According to the addition rule of exponents, whenever common bases raised to a certain exponent are multiplied, their powers can be added.

\(\displaystyle b\cdot b^{45}= b^1\cdot b^{45} = b^{1+45} = b^{46}\)

The answer is:  \(\displaystyle b^{46}\)

The quantity of the inner term is raised to the power of 25, which mean that the inner term will be multiplied by itself 25 times.

This means that we can multiply the inside power with the outside power as the exponential property.

\(\displaystyle (a^b)^c = a^{bc}\)

\(\displaystyle (3^{30})^{25} = 3^{30\times 25} = 3^{750}\)

The answer is:  \(\displaystyle 3^{750}\)

Use the product rule of exponents to simplify the exponents.

\(\displaystyle (a^b)^c =a^{bc}\)

Using this rule, we can rewrite the given problem.

\(\displaystyle 4(2^6)^{\frac{1}{3}} = 4(2^{6\times \frac{1}{3}}) =4(2^2)=4(4)=16\)

The answer is:  \(\displaystyle 16\)

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

\(\displaystyle (a^b)^c=a^{bc}\)

Simplify the problem.

\(\displaystyle -(2^3)^2= -(2^{3\times 2}) = -(2^6) = -64\)

The answer is:  \(\displaystyle -64\)

We will need to rewrite this in base two.

\(\displaystyle 4(32^9)^{10} = 2^2[(2^5)^9]^{10} = 2^2\cdot [(2^5)^9]^{10}\)

The exponents in the brackets and parentheses can be simplified by multiplication.  

\(\displaystyle 2^2\cdot [(2^5)^9]^{10} = 2^2\cdot 2^{450}\)

Add the exponents.

\(\displaystyle 2^2\cdot 2^{450} = 2^{2+450} = 2^{452}\)

The answer is:  \(\displaystyle 2^{452}\)