单项式的幂

Raise Normal Terms, Multiply Exponents

When working with complicated monomials (expressions that with a bunch of variables and numbers multiplied together, with no addition or subtraction), you might see something like this:

$$(2xy^4z^2)^3$$

This monomial has an exponent. This means you need to take the power of the whole monomial, internal exponents and all. This is called powers of monomials. The simplest way to do this is to:

1. Raise each term in the monomial to the power.
2. Simplify.

Let's try out the rules on the expression above.

A term taken to the power of $1$ is the same as the term itself. In other words, we can say that:

$$x = x^1$$

Let's rewrite the expression above so that each piece has its own exponent.

$$(2xy^4z^2)^3=( 2^1\ x^1\ y^4\ z^2 )\ ^3$$

(I've spaced out the pieces so that it's a little easier to see what's going on)

Next, we have to multiply each term inside the parentheses by the exponent outside the parentheses. This works similar to the distributive property, where we're distributing what is outside of the parentheses.

$$\begin{align} (2^1x^1y^4z^2)^{\color{#8925AE}3}&= 2^{1\cdot{\color{#8925AE}3}}\times x^{1\cdot{\color{#8925AE}3}}\times y^{4\cdot{\color{#8925AE}3}}\times z^{2\cdot{\color{#8925AE}3}} \end{align}$$

Finally, simplify. $$\begin{align} &2^{1\cdot{\color{#8925AE}3}}\times x^{1\cdot{\color{#8925AE}3}}\times y^{4\cdot{\color{#8925AE}3}}\times z^{2\cdot{\color{#8925AE}3}}\\ &=8x^3y^{12}z^6 \end{align}$$

Example Walk through $\\$

Taking a monomial to a power is not so different from taking a constant to a power. Start with a few warm up questions:

$$2^4=$$ $$(x^2)^4 =$$ $$(2x)^4=$$ $$(2x^2)^4=$$

Now, take a look at this one: $$(2x^3y^2)^4=$$

This expression looks more complicated than the warm up questions, but it is still manageable. Disregard the power outside of the parenthesis for a moment and treat each base as its own little unit: in this case, we have $2$, $x^3$ and $y^2$.

Then, take a look at the power outside of the parenthesis and distribute it to each unit we just identified. This simple breakdown will allow you to solve the powers of monomials!

Now, try a problem to test your understanding.

This system works when we're evaluating rational exponents as well! This method of evaluating the power of monomials will be really helpful when solving equations with rational values.

(Video) Powers of Monomials

by TeacherTube Math

This video by TeacherTube Math covers powers of monomials.

$\large{\text{Summary}}$

We can evaluate terms with exponents that are then raised to another power. Raise each piece of the term (coefficients and each individual variable) by that power. If the variable has an exponent, then you multiply the power of the expression by the power of the exponent. This is called the power of monomials.

Let's see this in action with an example. The first she looks at is: $$(x^2)^3$$ Here, we have a variable squared being raised to the third power. We simplify an exponent raised to a power by multiplying the powers. $$(x^2)^3=x^{2\cdot3}=x^6$$

The next example is: $$(2x^4)^6$$ We have a coefficient, $2$, and a variable raised to the fourth power. We simplify by raising each of these to the sixth power.

$$\begin{align} (2x^4)^6&=2^6\times x^{4^6}\\ &=64\times x^{4\cdot6}\\ &=64x^{24} \end{align}$$

The third example is: $$(3p^2r^3t)^3$$ This example has multiple variables. We raise each of these to the third power. $$\begin{align} (3p^2r^3t)^3&=3^3\times p^{2^3}\times r^{3^3}\times t^3\\ &=27\times p^{2\cdot3}\times r^{3\cdot3}\times t^3\\ &=27p^6r^9t^3 \end{align}$$