We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

\(\displaystyle 4=2^2\rightarrow 4^2=(2^2)^2\)

\(\displaystyle 16=2^4\rightarrow 16^3=(2^4)^3\)

\(\displaystyle 32=2^5\rightarrow 32^2=(2^5)^2\)

We can rewrite the original equation in these terms.

\(\displaystyle 2^n=(2^2)^2*(2^4)^3*(2^5)^2\)

Simplify exponents.

\(\displaystyle 2^n=(2^4)*(2^{12})*(2^{10})\)

Finally, combine terms.

\(\displaystyle 2^n=2^{4+12+10}=2^{26}\)

From this equation, we can see that \(\displaystyle n=26\).

Combining the powers, we get \(\displaystyle 1024=2^{x}\).

From here we can use logarithms, or simply guess and check to get \(\displaystyle x=10\).

When multiplying exponents with the same base, we use the rules of exponents.

This means you must simply add the exponents together as shown below:

\(\displaystyle x^{4}\cdot x^{5} = x^{4+5} = x^{9}\)

When you multiply exponents, you add the common bases:

y⁴ x⁷ + y⁵x⁶ + y¹⁸x⁴ + y³x²⁶

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

\(\displaystyle 3^{y-1}*3^2=27^y3^y\)

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

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

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

\(\displaystyle 3^{(y-1)+2}=(3)^{3y}*3^y\)

\(\displaystyle 3^{y+1}=3^{3y+y}=3^{4y}\)

We now know that the exponents must be equal, and can solve for \(\displaystyle y\).

\(\displaystyle y+1=4y\)

\(\displaystyle 1=3y\)

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

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.

To determine the value of this expression, it is not necessary to determine the values of each term's power.  Instead, since these powers have the same bases and are multiplied, the powers can be added.

\(\displaystyle 3^2\times3^3\times3^{-4} = 3^{2+3+(-4)} = 3^1=3\)

The answer is \(\displaystyle 3\).

Multiplying two exponents that have the same base is the equivalent of simply adding the exponents.

So \(\displaystyle y = 5^{a} * 5^{b}\) is the same as \(\displaystyle y = 5^{a + b}\), and if \(\displaystyle a + b = 3\), then \(\displaystyle y = 5^{3}\) or \(\displaystyle 125\). 

The exponents cannot be added unless the both bases are alike and similar bases must be multiplied with each other.  Rewrite the nine with a base of three.

\(\displaystyle 9=3^2\)

Rewrite the expression.  

\(\displaystyle 3^x+9^x= 3^x+3^{2x}\)

Do not add the exponents, since similar bases are added and are not multiplied with each other!

The answer is: \(\displaystyle 3^x+3^{2x}\)

When we multiply two polynomials with exponents, we add their exponents together. Therefore,

\(\displaystyle x^5y^4z^2 \cdot x^4y^5z^2 = x^9y^9z^4\)