Outro W14

9.7 Higher Roots

• Properties of
• $\sqrt[n]{a}$ when n is an even number and
  • a≥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≥2, when n is odd $\sqrt[n]{a^n}=a$
  • For any integer n≥2, when n is even $\sqrt[n]{a^n}=\mid a\mid$
• $\sqrt[n]{a}$ is considered simplified if a has no factors of $m^n$.
• Product Property of nth Roots
  
  \(\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}\:\:\:\:\:and\:\:\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}\)
• Quotient Property of nth Roots
  
  \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\:\:\:\:\:\:and\:\:\:\:\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
• 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 $a^m\cdot a^n=a^{m+n}$
  • Power Property $\left(a^m\right)^n=a^{m\cdot n}$
  • Product to a Power $\left(ab\right)^m=a^m\cdot b^m$
  • Quotient Property:
    
    \(\frac{a^m}{a^n}=a^{m-n}\:,\:\:a\ne0\:\:\)
  • Zero Exponent Definition $a^0=1$
  • Quotient to a Power Property $\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m},\:\:\:b\ne0$

7.6 Quadratic Equations

• Zero Product Property If a⋅b=0, then either a=0 or b=0 or both. See Example 7.69 Links to an external site..
• Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See Example 7.73 Links to an external site..
  1. Step 1. Write the quadratic equation in standard form, $ax^2+bx+c=0$.
  2. Step 2. Factor the quadratic expression.
  3. Step 3. Use the Zero Product Property.
  4. Step 4. Solve the linear equations.
  5. Step 5. Check.
• Use a problem solving strategy to solve word problems See Example 7.80 Links to an external site..
  1. Step 1. Read the problem. Make sure all the words and ideas are understood.
  2. Step 2. Identify what we are looking for.
  3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
  4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
  5. Step 5. Solve the equation using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.

10.1 Solve Quadratic Equations Using the Square Root Property

• Square Root Property
  If $x^2=k$, and k≥0, then $x=\sqrt{k}\:\:or\:\:x=-\sqrt{k}$