Lesson 1.2: Exponents and Scientific Notation

New Concept

This lesson introduces the fundamental rules of exponents, such as the product, quotient, and power rules. Mastering these will allow you to simplify complex algebraic expressions and efficiently work with scientific notation for very large and small numbers.

What’s next

You'll start by exploring the core rules of exponents through interactive examples and practice cards to build your foundational skills for the upcoming challenges.

Property

For any real number $a$ and natural numbers $m$ and $n$, the product rule of exponents states that

Examples

• To simplify $g^4 \cdot g^2$, we add the exponents: $g^{4+2} = g^6$.
• To simplify $(-5)^4 \cdot (-5)$, we recognize that $(-5)$ is $(-5)^1$. So, we have $(-5)^{4+1} = (-5)^5$.
• We can combine multiple terms: $y^3 \cdot y^6 \cdot y^2 = y^{3+6+2} = y^{11}$.

Explanation

When multiplying terms with the same base, keep the base and add the exponents. This is a shortcut for counting all the individual factors. For example, $x^2 \cdot x^3$ means $(x \cdot x) \cdot (x \cdot x \cdot x)$, which is simply $x^5$.

Property

For any real number $a$ and natural numbers $m$ and $n$, such that $m > n$, the quotient rule of exponents states that

Examples

• To simplify $\dfrac{(-7)^{10}}{(-7)^6}$, we subtract the exponents: $(-7)^{10-6} = (-7)^4$.
• To simplify $\dfrac{p^{15}}{p^8}$, we subtract the exponents: $p^{15-8} = p^7$.
• To simplify $\dfrac{(x\sqrt{5})^7}{x\sqrt{5}}$, the denominator has an exponent of 1: $(x\sqrt{5})^{7-1} = (x\sqrt{5})^6$.

Explanation

When dividing terms that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. This rule is a fast way to cancel out common factors between the top and bottom of a fraction.

Property

For any real number $a$ and positive integers $m$ and $n$, the power rule of exponents states that

Examples

• To simplify $(y^3)^5$, we multiply the exponents: $y^{3 \cdot 5} = y^{15}$.
• To simplify $((3k)^4)^2$, the entire base $(3k)$ is raised to the product of the powers: $(3k)^{4 \cdot 2} = (3k)^8$.
• To simplify $((-5)^2)^6$, we multiply the exponents: $(-5)^{2 \cdot 6} = (-5)^{12}$.

Explanation

When an exponential expression is raised to another power, you multiply the exponents. Think of it as repeated groups. For instance, $(x^3)^2$ means two groups of $x^3$, giving you $x^6$. Just multiply the powers: $3 \times 2 = 6$.

Property

For any nonzero real number $a$, the zero exponent rule of exponents states that

Examples

• To simplify $\dfrac{k^7}{k^7}$, we can use the quotient rule: $k^{7-7} = k^0 = 1$.
• To simplify $\dfrac{10y^2}{y^2}$, we can separate the coefficient: $10 \cdot \dfrac{y^2}{y^2} = 10 \cdot y^0 = 10 \cdot 1 = 10$.
• To simplify $\dfrac{-4(m^3)^5}{(m^3)^5}$, we get $-4 \cdot (m^3)^{5-5} = -4 \cdot (m^3)^0 = -4 \cdot 1 = -4$.

Explanation

Any nonzero number or variable raised to the power of zero always equals 1. This rule comes from the fact that any number divided by itself is 1. For example, $\dfrac{x^5}{x^5} = 1$, and by the quotient rule, it's also $x^{5-5} = x^0$.

Property

For any nonzero real number $a$ and natural number $n$, the negative rule of exponents states that

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

Examples

• To simplify $\dfrac{m^5}{m^{12}}$, we subtract exponents to get $m^{5-12} = m^{-7}$, which is written with a positive exponent as $\dfrac{1}{m^7}$.
• To simplify $k^4 \cdot k^{-9}$, we add exponents to get $k^{4+(-9)} = k^{-5}$, which becomes $\dfrac{1}{k^5}$.
• To simplify $\dfrac{(2y)^3}{(2y)^9}$, we get $(2y)^{3-9} = (2y)^{-6}$, which becomes $\dfrac{1}{(2y)^6}$.

Explanation

A negative exponent means you should take the reciprocal of the base. Think of it as an instruction to move the term across the fraction bar. For example, $x^{-2}$ moves to the denominator to become $\dfrac{1}{x^2}$.

Property

For any real numbers $a$ and $b$ and any integer $n$, the power of a product rule of exponents states that

For any real numbers $a$ and $b$ and any integer $n$, the power of a quotient rule of exponents states that

Examples

• To simplify $(xy^3)^4$, distribute the exponent 4 to both $x$ and $y^3$: $x^4(y^3)^4 = x^4y^{12}$.
• To simplify $\left(\dfrac{2}{k^5}\right)^4$, distribute the exponent 4 to the numerator and denominator: $\dfrac{2^4}{(k^5)^4} = \dfrac{16}{k^{20}}$.
• To simplify $(m^4n^{-3})^2$, distribute the exponent 2 to both factors: $(m^4)^2(n^{-3})^2 = m^8n^{-6} = \dfrac{m^8}{n^6}$.

Explanation

When a product or a quotient is raised to a power, you can 'distribute' that power to each factor inside. This rule works because the exponent applies to everything being multiplied or divided within the parentheses.

Property

A number is written in scientific notation if it is written in the form $a \times 10^n$, where $1 \le |a| < 10$ and $n$ is an integer.
To write a number in scientific notation, move the decimal point to the right of the first digit.
The number of places you moved the decimal point is the exponent $n$. If you moved the decimal left, $n$ is positive.
If you moved the decimal right, $n$ is negative.

Examples

• To write 5,820,000 in scientific notation, move the decimal 6 places to the left to get 5.82. Since you moved left, the exponent is positive: $5.82 \times 10^6$.
• To write 0.00071 in scientific notation, move the decimal 4 places to the right to get 7.1. Since you moved right, the exponent is negative: $7.1 \times 10^{-4}$.
• To convert $2.9 \times 10^{-5}$ to standard notation, move the decimal 5 places to the left because the exponent is negative: $0.000029$.

Explanation

Scientific notation is a compact way to write very large or tiny numbers. It has two parts: a number between 1 and 10, and a power of 10. The exponent tells you how many places the decimal point moved to get there.