Although we have different bases, there is a commonality between base \(\displaystyle 4\) and \(\displaystyle 8\).

We can convert them all to base \(\displaystyle 2\). 

\(\displaystyle 4=2^2\) and \(\displaystyle 8=2^3\)

Now, let's figure out the exponents after we converted the factors to base \(\displaystyle 2\) by creating proportions.

 The top of the right fraction represents exponent of base \(\displaystyle 4\). The bottom of the left fraction represents exponent of base \(\displaystyle 2\). 

\(\displaystyle \frac{1}{2}=\frac{9}{x}\)

When we cross-multiply we get \(\displaystyle x=18\). 

 The top of the right fraction represents exponent of base \(\displaystyle 8\). The bottom of the left fraction represents exponent of base \(\displaystyle 2\). 

\(\displaystyle \frac{1}{3}=\frac{4}{x}\)

When we cross-multiply we get \(\displaystyle x=12\). 

We now have the same base in the factors: \(\displaystyle 2^{18}\) and \(\displaystyle 2^{12}\). 

With same bases, we can subtract the exponents.

\(\displaystyle \frac{2^{18}}{2^{12}}=2^{18-12}\)

\(\displaystyle 2^{18-12}=2^6\)

When dividing exponents, we must first check to see if the factors have the same base. If they have the same bases, then we can subtract the exponents.

Let's work from left to right.

On the left is:

 \(\displaystyle \frac{x^4}{x^{3}}=x^{4-3}\)

\(\displaystyle x^{4-3}=x\)

The numerator of the fraction had a higher exponent value for \(\displaystyle x\); therefore, \(\displaystyle x\) remains in the numerator.

Next, we have: 

\(\displaystyle \frac{y^8}{y^9}=y^{8-9}\)

Remember that when you subtract a larger number from a smaller number the answer is negative.  Since \(\displaystyle 9\) is greater than \(\displaystyle 8\) and is after the subtraction sign, our answer is negative.

\(\displaystyle y^{8-9}=y^{-1}\)

The denominator of the fraction had a higher exponent value for \(\displaystyle y\); therefore, \(\displaystyle y\) remains in the denominator.

Last, we have:

 \(\displaystyle \frac{z^5}{z^8}=z^{5-8}\)

Remember that when you subtract a larger number from a smaller number the answer is negative.  Since \(\displaystyle 8\) is greater than \(\displaystyle 5\) and is after the subtraction sign, our answer is negative.

\(\displaystyle z^{5-8}=z^{-3}\)

The denominator of the fraction had a higher exponent value for \(\displaystyle z\); therefore, \(\displaystyle z\) remains in the denominator.

Now we can place all of the terms into one equation: \(\displaystyle xy^{-1}z^{-3}\).

Negative exponents are dropped to the bottom of the fraction and their sign is eliminated.

Therefore, our final answer becomes:

 \(\displaystyle \frac{x}{yz^3}\)

Remember, with negative powers, it means they are in the denominator and the exponent becomes positive. 

When multiplying exponents, we must first check to see if the factors have the same base. If they have the same bases, then we can add the exponents.

These factors have different bases; therefore, we cannot simplify any further.  

The answer is as follows:

\(\displaystyle x^7*y^6\)

Although they have different bases, they do have the same exponent. We can essentially multiply the base but keep the exponent constant.

\(\displaystyle 2^7*3^7=(2*3)^7\)

The answer is as follows:

\(\displaystyle (2*3)^7=6^7\)

Although we have different bases, there is a commonality between base \(\displaystyle 27\) and \(\displaystyle 9\).

We can convert them all to base \(\displaystyle 3\). 

\(\displaystyle 27=3^3\) and \(\displaystyle 9=3^2\) 

Now, let's figure out the exponents after we converted the factors to base \(\displaystyle 3\) by creating proportions.

The top of the right fraction represents the exponent of base \(\displaystyle 27\). The bottom of the left fraction represents the exponent of base \(\displaystyle 3\).

\(\displaystyle \frac{1}{3}=\frac{3}{x}\)

When we cross-multiply we get \(\displaystyle x=9\). 

The top of the right fracrion represents the exponent of base \(\displaystyle 9\). The bottom of the left fraction represents the exponent of base \(\displaystyle 3\).

\(\displaystyle \frac{1}{2}=\frac{4}{x}\)

When we cross-multiply we get \(\displaystyle x=8\). 

We now have the same base in the factors: \(\displaystyle 3^{9}\) and \(\displaystyle 3^{8}\).

With same bases, we can add the exponents.

\(\displaystyle 3^9*3^{8}=3^{9+8}\)

\(\displaystyle 3^{9+8}=3^{17}\)

This problem can be solved using the properties of exponents. When dividing exponents, you subtract the numerator from the denominator. Remember that a varaible with a negative exponent is the reciprocal of one with a positive exponent (i.e. \(\displaystyle a^{-1}=\frac{1}{a}\)).

\(\displaystyle \frac{a^2b^7x^6y}{yza^7b^7}=\frac{a^{2}b^{7}x^{6}y}{a^{7}b^{7}yz}\)

\(\displaystyle \frac{a^{2}b^{7}x^{6}y}{a^{7}b^{7}yz}=\frac{a^{-5}b^{0}x^{6}y^{0}}{z}\)

\(\displaystyle \frac{a^{-5}b^{0}x^{6}y^{0}}{z}=\frac{a^{-5}x^{6}}{z}\)

Remember that a varaible with a negative exponent is the reciprocal of one with a positive exponent (i.e. \(\displaystyle a^{-1}=\frac{1}{a}\)).

\(\displaystyle \frac{a^{-5}x^{6}}{z}=\frac{x^{6}}{a^{5}z}\)

To simplify this expression you will need to know the multiplication rule of exponents \(\displaystyle a^m*a^n=a^{m+n}\), the division rule of exponents \(\displaystyle \frac{a^m}{a^n}=a^{m-n}\), the power rule \(\displaystyle (a^m)^n=a^{m*n}\), and the fact that any exponent to the zero power is equal to 1.

\(\displaystyle \frac{27x^5y^{10}q^4}{9xy^{10}q}*15[x^4q^3]^2\)

Start by simplifying the left side:

\(\displaystyle 3x^4q^3*15[x^4q^3]^2\)

Now use the power rule:

\(\displaystyle 3x^4q^3*15x^8q^6\)

Multiplication rule:

\(\displaystyle 45x^{12}q^9\) is the final answer

Note that \(\displaystyle \frac{y^{10}}{y^{10}}=y^0=1\); there are several ways to visualize this, but the quickest and simplest is to remind yourself that 567/567=1 and so does 1,345,269/1,345,269. Any value over that same value equals 1. 

Use the rules of exponent division to cancel out like-terms:

\(\displaystyle \frac{4x^{2}y^{6}z^{3}}{2x^{3}y^{3}z^{6}}\)

\(\displaystyle =2x^{-1}y^{3}z^{-3}\)

\(\displaystyle =\frac{2y^{3}}{xz^{3}}\)

Simplify the following expression:

\(\displaystyle 5x^6*8x^3\)

When multiplying exponents, we need to recall the rule. When we multiply exponents of like base together, we add the exponents. This is seperate from the numbers out in front (the 5 and the 8) those two numbers will be multiplied as normal.

We can sort of rearrange the expression to get the following:

\(\displaystyle 5x^6*8x^3=(5*8)(x^6*x^3)=40x^{6+3}=40x^9\)

Making our answer:

\(\displaystyle 40x^9\)

Simplify the following expression:

\(\displaystyle \frac{343x^6y^3b^7}{49x^4y^2b^9}\)

Recall that when dividing exponents of like base, we need to subtract the exponents to simplify.

For our integers, we can either treat them like regular division, or rewrite them as exponents:

\(\displaystyle \frac{7^3x^6y^3b^7}{7^2x^4y^2b^9}=7^{3-2}x^{6-4}y^{3-2}b^{7-9}=7x^2yb^{-2}=\frac{7x^2y}{b^2}\)

So our answer becomes:

\(\displaystyle \frac{7x^2y}{b^2}\)