Key Concepts

• Exponential Notation

• 
  This is read $a$ to the ${m}^{\mathrm{th}}$ power.

• Properties of Exponent
  • If $a,b$ are real numbers and $m,n$ are integers, then
    $\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \mathbf{\text{Power of a Product Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & {\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,a\ne 0\hfill \\ \mathbf{\text{Power of a Quotient Property}}\hfill & & & {\left({\Large\frac{a}{b}}\right)}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}={\Large\frac{1}{{a}^{n}}}\hfill \end{array}$
• Convert from Decimal Notation to Scientific Notation: 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. Write the number as a product with a power of $10$.
     • If the original number is greater than $1$, the power of $10$ will be ${10}^{n}$ .
     • If the original number is between $0$ and $1$, the power of $10$ will be ${10}^{n}$ .
  4. Check.
• Convert from Scientific Notation to Decimal Notation: To convert scientific notation to decimal form:
  1. Determine the exponent, $n$ , on the factor $10$.
  2. Move the decimal $n$ places, adding zeros if needed.
     • If the exponent is positive, move the decimal point $n$ places to the right.
     • If the exponent is negative, move the decimal point $|n|$ places to the left.
  3. Check.
• Square Root Notation $\sqrt{m}$ is read ‘the square root of $m$ ’.  If $m={n}^{2}$ , then $\sqrt{m}=n$ , for $n\ge 0$ .
  
• Square Roots and Area If the area of the square is A square units, the length of a side is $\sqrt{A}$ units.
• Square Roots and Gravity 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 evaluating the expression ${\Large\frac{\sqrt{h}}{4}}$.
• Square Roots and Accident Investigations Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is $d$ feet, then the speed of the car can be found by evaluating $\sqrt{24d}$.
  .
• Use a strategy for applications with square roots.
  • Identify what you are asked to find.
  • Write a phrase that gives the information to find it.
  • Translate the phrase to an expression.
  • Simplify the expression.
  • Write a complete sentence that answers the question.

Glossary

negative exponent

If $n$ is a positive integer and $a\ne 0$ , then ${a}^{-n}=\frac{1}{{a}^{n}}$ .

scientific notation

A number expressed in the form $a\times {10}^{n}$, where $a\ge 1$ and $a<10$, and $n$ is an integer.

perfect square

A perfect square is the square of a whole number.

square root of a number

A number whose square is $m$ is called a square root of $m$.
If ${n}^{2}=m$, then $n$ is a square root of $m$.

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