Lesson 5.2: Properties of Exponents and Scientific Notation

New Concept

In this lesson, you'll learn the fundamental properties of exponents. These rules are your toolkit for simplifying complex algebraic expressions and for writing very large or small numbers using scientific notation, such as $a \times 10^n$.

What’s next

You're ready to start! Next, you'll work through interactive examples and practice cards to master each exponent property one by one.

Property

If $a$ is a real number and $m$ and $n$ are integers, then

To multiply with like bases, add the exponents.

Examples

• To simplify $z^4 \cdot z^7$, you keep the base $z$ and add the exponents: $z^{4+7} = z^{11}$.
• In $3a^5 \cdot 5a^2$, multiply the coefficients and add the exponents of the common base: $(3 \cdot 5)a^{5+2} = 15a^7$.
• For $p^3 \cdot p^6 \cdot p^2$, since all bases are the same, you add all the exponents: $p^{3+6+2} = p^{11}$.

Explanation

When you multiply powers with the same base, you're just combining their factors. For example, $x^2 \cdot x^3$ is $(x \cdot x) \cdot (x \cdot x \cdot x)$, which is five $x$'s multiplied together, or $x^5$. Just add the exponents!

Property

If $a$ is a real number, $a \neq 0$, and $m$ and $n$ are integers, then

and

Examples

• To simplify $\frac{y^{10}}{y^4}$, subtract the exponents since the larger exponent is in the numerator: $y^{10-4} = y^6$.
• In $\frac{5^{12}}{5^3}$, you use the same rule: $5^{12-3} = 5^9$.
• For $\frac{c^5}{c^{15}}$, the larger exponent is in the denominator, so the result is a fraction: $\frac{1}{c^{15-5}} = \frac{1}{c^{10}}$.

Explanation

Dividing powers with the same base is like canceling out common factors. If you have more factors on top (like $\frac{x^5}{x^2}$), the leftovers stay on top ($x^3$). If more are on the bottom, the leftovers stay below.

Property

Zero Exponent Property
If $a$ is a non-zero number, then $a^0 = 1$.

Properties of Negative Exponents
If $n$ is an integer and $a \neq 0$, then $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$.

Quotient to a Negative Power Property
If $a$ and $b$ are real numbers, $a \neq 0$, $b \neq 0$ and $n$ is an integer, then

Property

Power Property for Exponents
If $a$ is a real number and $m$ and $n$ are integers, then

Product to a Power Property for Exponents
If $a$ and $b$ are real numbers and $m$ is a whole number, then

Quotient to a Power Property for Exponents
If $a$ and $b$ are real numbers, $b \neq 0$, and $m$ is an integer, then

Property

A number is expressed in scientific notation when it is of the form

where $1 \leq |a| < 10$ and $n$ is an integer.

How To Convert a Decimal to Scientific Notation:

1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$, that the decimal point was moved.
3. If the original number is greater than 1, the power of 10 will be $10^n$. If it's between 0 and 1, the power will be $10^{-n}$.

Examples

• To write 5,280,000 in scientific notation, move the decimal 6 places to the left, which gives $5.28 \times 10^6$.
• To convert $3.1 \times 10^{-4}$ to decimal form, move the decimal 4 places to the left, resulting in $0.00031$.
• To multiply $(2 \times 10^4)(3 \times 10^2)$, multiply the numbers ($2 \cdot 3=6$) and add the exponents ($10^{4+2}=10^6$), giving $6 \times 10^6$.