Lesson 6: Chapter Summary and Review

New Concept

This lesson connects the laws of exponents with the properties of radicals. You'll learn to simplify expressions involving powers like $a^n$ and roots like $\sqrt{a}$, preparing you to solve complex equations and understand their real-world applications.

What’s next

Now, you will solidify these concepts through a series of practice cards and challenge problems that test your skills on both exponents and radicals.

Property

First Law of Exponents

Second Law of Exponents

Property

Zero as an Exponent

Negative Exponents

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Property

A number is in scientific notation if it is expressed as the product of a number between 1 and 10 and a power of 10. To write a number in scientific notation, we first position the decimal point and then determine the correct power of 10.

Examples

• The large number 8,120,000 is written as $8.12 \times 10^6$ because the decimal point moves 6 places to the left.
• The small number 0.000059 is written as $5.9 \times 10^{-5}$ because the decimal point moves 5 places to the right.
• To convert $642 \times 10^3$ to scientific notation, first write 642 as $6.42 \times 10^2$, then combine the powers: $(6.42 \times 10^2) \times 10^3 = 6.42 \times 10^5$.

Explanation

Scientific notation is a compact and standard way to write very large or very small numbers. It makes them easier to read, compare, and use in calculations without managing long strings of zeros.

Property

Product Rule for Radicals
If $a, b \geq 0$, then $\sqrt{ab} = \sqrt{a}\sqrt{b}$

Quotient Rule for Radicals
If $a \geq 0, b > 0$ then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

It is just as important to remember that we do not have a sum or difference rule for radicals. That is, in general, $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.

Property

To Simplify a Square Root

1. Factor any perfect squares from the radicand.
2. Use the product rule to write the radical as a product of two square roots.
3. Simplify the square root of the perfect square.

Square roots with identical radicands are called like radicals. We can add or subtract like radicals by adding or subtracting their coefficients.

Examples

• To simplify $\sqrt{72}$, find the largest perfect square factor: $\sqrt{36 \cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$.
• To combine like radicals, treat the radical part as a variable: $8\sqrt{3} + 3\sqrt{3} = (8+3)\sqrt{3} = 11\sqrt{3}$.
• To combine $\sqrt{20} + \sqrt{45}$, first simplify each one: $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$.

Property

To Solve a Radical Equation

1. Isolate the radical on one side of the equation.
2. Square both sides of the equation.
3. Continue as usual to solve for the variable.

The technique of squaring both sides may introduce extraneous solutions, which are solutions that do not work in the original equation. Therefore, you must check your final answers.

Examples

• To solve $\sqrt{x} = 8$, square both sides: $(\sqrt{x})^2 = 8^2$, which gives $x = 64$. The check, $\sqrt{64}=8$, works.
• To solve $\sqrt{y-2} + 5 = 9$, first isolate the radical to get $\sqrt{y-2} = 4$. Squaring both sides gives $y-2 = 16$, so $y = 18$.
• To solve $\sqrt{z} = -6$, squaring both sides gives $z = 36$. However, checking this in the original equation gives $\sqrt{36} = 6 \neq -6$. Therefore, there is no solution.