### Key Concepts

### 9.1 Simplify and Use Square Roots

- Note that the square root of a negative number is not a real number.
- Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
- We can estimate square roots using nearby perfect squares.
- We can approximate square roots using a calculator.
- When we use the radical sign to take the square root of a variable expression, we should specify that $x\ge 0$ to make sure we get the principal square root.

### 9.2 Simplify Square Roots

**Simplified Square Root**$\sqrt{a}$ is considered simplified if $a$ has no perfect-square factors.**Product Property of Square Roots**If*a*,*b*are non-negative real numbers, then

$$\sqrt{ab}=\sqrt{a}\xb7\sqrt{b}$$**Simplify a Square Root Using the Product Property**To simplify a square root using the Product Property:- Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
- Step 2. Use the product rule to rewrite the radical as the product of two radicals.
- Step 3. Simplify the square root of the perfect square.

**Quotient Property of Square Roots**If*a*,*b*are non-negative real numbers and $b\ne 0$, then

$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$$**Simplify a Square Root Using the Quotient Property**To simplify a square root using the Quotient Property:- Step 1. Simplify the fraction in the radicand, if possible.
- Step 2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
- Step 3. Simplify the radicals in the numerator and the denominator.

### 9.3 Add and Subtract Square Roots

- To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
- Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

### 9.4 Multiply Square Roots

**Product Property of Square Roots**If*a*,*b*are nonnegative real numbers, then

$$\sqrt{ab}=\sqrt{a}\xb7\sqrt{b}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sqrt{a}\xb7\sqrt{b}=\sqrt{ab}$$**Special formulas**for multiplying binomials and conjugates:

$$\begin{array}{cccc}{(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\hfill & & & (a-b)(a+b)={a}^{2}-{b}^{2}\hfill \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\hfill & & & \end{array}$$- The FOIL method can be used to multiply binomials containing radicals.

### 9.5 Divide Square Roots

**Quotient Property of Square Roots**- If
*a*,*b*are non-negative real numbers and $b\ne 0$, then$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$$

- If
**Simplified Square Roots**

A square root is considered simplified if there are- no perfect square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction

### 9.6 Solve Equations with Square Roots

**To Solve a Radical Equation:**- Step 1. Isolate the radical on one side of the equation.
- Step 2. Square both sides of the equation.
- Step 3. Solve the new equation.
- Step 4. Check the answer. Some solutions obtained may not work in the original equation.

**Solving Applications with Formulas**- Step 1.
**Read**the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for by choosing a variable to represent it. - Step 4.
**Translate**into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. - Step 5.
**Solve the equation**using good algebra techniques. - Step 6.
**Check**the answer in the problem and make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.
**Area of a Square**

**Falling Objects**- On Earth, if an object is dropped from a height of $h$ feet, the time in seconds it will take to reach the ground is found by using the formula $t=\frac{\sqrt{h}}{4}$.

**Skid Marks and Speed of a Car**- If the length of the skid marks is
*d*feet, then the speed,*s*, of the car before the brakes were applied can be found by using the formula $s=\sqrt{24d}$.

- If the length of the skid marks is

### 9.7 Higher Roots

**Properties of**- $\sqrt[n]{a}$ when $n$ is an even number and
- $a\ge 0$, then $\sqrt[n]{a}$ is a real number
- $a<0$, then $\sqrt[n]{a}$ is not a real number
- When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of
*a*. - For any integer $n\ge 2$, when
*n*is odd $\sqrt[n]{{a}^{n}}=a$ - For any integer $n\ge 2$, when
*n*is even $\sqrt[n]{{a}^{n}}=\left|a\right|$

- $\sqrt[n]{a}$ is considered simplified if
*a*has no factors of ${m}^{n}$. **Product Property of***n*th Roots

$$\sqrt[n]{ab}=\sqrt[n]{a}\xb7\sqrt[n]{b}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\sqrt[n]{a}\xb7\sqrt[n]{b}=\sqrt[n]{ab}$$**Quotient Property of***n*th Roots

$$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$$- To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

### 9.8 Rational Exponents

**Summary of Exponent Properties**- If $a,b$ are real numbers and $m,n$ are rational numbers, then
**Product Property**${a}^{m}\xb7{a}^{n}={a}^{m+n}$**Power Property**${\left({a}^{m}\right)}^{n}={a}^{m\xb7n}$**Product to a Power**${\left(ab\right)}^{m}={a}^{m}{b}^{m}$**Quotient Property**:

$$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}m>n$$$$\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}n>m$$**Zero Exponent Definition**${a}^{0}=1$, $a\ne 0$**Quotient to a Power Property**${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.5em}{0ex}}b\ne 0$