Question

How do you simplify $2{y^2} \times 3x$ and write it using only positive exponents?

Answer 1

Hint: The given simplification is $2{y^2} \times 3x$
Let’s take the given equation $2{y^2} \times 3x$
We use the positive exponents
First we separate the given information into variables and constants.
After that we multiply the constants, we get the number. And then multiply the two variables, we get the variables. After that we explain the positive exponential function.
And then finally we get the result.

Complete step by step answer:
The given simplification is $2{y^2} \times 3x$
Let’s take the given equation $2{y^2} \times 3x$
First we factor the variables and constants, hence we get
$ \Rightarrow (2 \times 3)({y^2} \times x)$
Multiply$2$by$3$, hence we get
$ \Rightarrow 6({y^2} \times x)$
Now multiply the two variables, hence we get
$ \Rightarrow 6{y^2}x$
And then finally we get the result.
The exponential function $x \to {e^x}$ has a fundamental property involving products:
For every $x,y \in \mathbb{R},{e^{x + y}} = {e^x} \times {e^y}$
So, exponential function transforms sums into products. Of course, you can write,
$ \Rightarrow {e^{{x_1} + {x_2} + \ldots + {x_n}}} = {e^{{x_1}}} \times {e^{{x_2}}} \times \ldots \times {e^{{x_n}}}$
For any numbers${x_1}, \ldots ,{x_n}$
Note that there exist other exponential functions: $x \to {10^x},x \to {2^x}, \ldots x \to {a^x}$ for any positive real $a$. All of them have the same property involving products.

Note: Exponential notation is a way of shorthand for very large numbers and very small numbers. They are the number that you see at the top right of another number, called the base, as in${10^2}$, where the$10$is the base and$2$is the exponent. The exponent tells you how many times you multiply the base with itself:
$ \Rightarrow {10^2} = 10 \times 10 = 100$
This goes for any number:
$ \Rightarrow {2^4} = 2 \times 2 \times 2 \times 2 = 16$
$ \Rightarrow {10^5} = 10 \times 10 \times 10 \times 10 \times 10$
So, ${10^5}$ is a short way of writing a $1$ with $5$ zeroes! This will come in handy if we deal with really large numbers.