Since the base is 5 for each term, we can say 2 + n =12.  Solve the equation for n by subtracting 2 from both sides to get n = 10.

Start by simplifying each individual term between the plus signs. We can add the exponents in \(\displaystyle x^3x^6\) and \(\displaystyle x^3x^2x^2x^2\) so each of those terms becomes \(\displaystyle x^9\). Then multiply the exponents in \(\displaystyle (x^3)^3\) so that term also becomes \(\displaystyle x^9\). Thus, we have simplified the expression to \(\displaystyle x^9 + x^9 + x^9\) which is \(\displaystyle 3x^{9}\).

First, simplify \(\displaystyle x^{3}x^{5}\) by adding the exponents to get \(\displaystyle x^{8}\).

Then simplify \(\displaystyle (y^{2})^{2}\) by multiplying the exponents to get \(\displaystyle y^{4}\).

This gives us \(\displaystyle x^{8} - x^{^{2}} + y^{4}\). We cannot simplify any further.

To attempt this problem, note that \(\displaystyle 3^2=(3)^2=9^1=9\).

Now note that when multiplying numbers, if the base is the same, we may add the exponents:

\(\displaystyle 3^4 \cdot 3^8 = 3^{4+8}=3^{12}\)

This can in turn be written in terms of nine as follows (recall above)

\(\displaystyle 3^{12}=9^{\frac{12}{2}}=9^6\)

\(\displaystyle 9^6=9^x\)

\(\displaystyle x=6\)

When dealing with exponenents, when multiplying two like bases together, add their exponents:

\(\displaystyle 3^2\cdot3^5=3^7\)

However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:

\(\displaystyle (3^7)^3=3^{7\cdot 3}=3^{21}\)

\(\displaystyle 3^{21}=3^{x}\)

\(\displaystyle x=21\)

Begin by noting that the group (4²³ - 4²¹) has a common factor, namely 4²¹.  You can treat this like any other constant or variable and factor it out.  That would give you: 4²¹(4² - 1). Therefore, we know that:

3² * (4²³ - 4²¹) = 3² * 4²¹(4² - 1)

Now, 4² - 1 = 16 - 1 = 15 = 5 * 3.  Replace that in the original:

3² * 4²¹(4² - 1) = 3² * 4²¹(3 * 5)

Combining multiples withe same base, you get:

3³ * 4²¹ * 5

We can solve this without actually doing the math. Let's look at 6⁴ vs 5². 6⁴ is clearly bigger. Now let's look at 3² vs 4². 3² is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.

The first thing to note is the relationship between (–b) and (1 – b):

 (–b) < (1 – b) because (–b) + 1 = (1 – b).

Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.

Raising a negative number to an odd power produces another negative number.

Thus (–b)^a < (1 – b)^a < 0.

Simplifying the equation gives b⁶/(b^3+x) = b⁵.  

In order to satisfy this case, x must be equal to –2.

7/8 is being raised to the 5th power and to the ¹/₂ power at the same time. We multiply these to find n.