Notação cientica: Elevar uma raíz a potência

(Video) Raising Scientific Notation to a Power

by TN Learn

TN Learn has a good video that explains raising a number in scientific notation to a power. This topic combines order of operations with rules of exponents.

$\large{\text{Summary}}$

The gist of it is to take each part of the number individually. For example: $$(a\times 10^m)^n = (a^n\times 10^{m\cdot n})$$

We are in essence distributing the exponent. This somewhat resembles the distributive property.

Remember to always put your answer back into scientific notation. In scientific notation, $a\times 10^n$, $1\leq a < 10$.

Let's work through an example:

$$\text{What is } (3 \times 10^4)^2 \text{?}$$

We can start by applying the exponent to both parts. This is using the power rule for exponents on both the $3$ and the $10^4$.

$$(3 \times 10^4)^2 = 3^2 \times (10^4)^2$$

Now we can apply the exponents. Remember that raising a power to a power is the same as multiplying the powers.

$$3^2 \times (10^4)^2 =\boxed{9 \times 10^8}$$

Raising Constants, Monomials, and Scientific Notation to a Power

Raising a scientific notation to a power is similar to raising a number to a power. For example:

$$(3^2)^4 = 3^{3\times 4} = 3^{12}$$

We apply the same principle, but apply it to both the number and the power of ten. This is similar to raising a monomial to a power: $$(2x^3)^4 = 2^4x^{3\times 4} = 16x^{12}$$

Now, let's try to raise a scientific notation to a power: $$\left(2\times10^6\right)^{^3}=2^3\times\left(10^6\right)^3=8\times10^{3\times6}=8\times10^{18}$$

Try the following problem to test you understanding.

$$\textbf{Raising to a Power in Scientific Notation}$$

Example Walk Through

$$\large{\large{(2 \times 10^6)^3}}$$

Take at look at the expression above, how do we raise a scientific notation to a power?

This is similar to raising a number or a monomial to a power. In this case, we "distribute" $3$ to each component of the scientific notation. $$\large{(2^3) \times (10^6)^3}$$ Then, we solve! $$\begin{align}(2^3) \times (10^6)^3 &= 8 \times (10^6)^3\\ &= \boxed{8 \times 10^{18}} \end{align}$$