Key Concepts

• The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root.
•  A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex]\sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd, and [latex]\sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.
•  A radical can be expressed as an expression with a fractional exponent by following the convention [latex]\sqrt[n]{{{a}^{m}}}={{a}^{\frac{m}{n}}}[/latex]. Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions
• To find the prime factorization of a number, use the factor tree method to divide away small prime numbers until only prime factors remain. The product of those prime factors is the prime factorization of the number.
• To find the least common multiple (LCM) of a list of numbers, write each number as a product of its prime factors, select the largest instance of each prime that appears in any one number, then multiply the selections together to obtain the LCM.
• To add or subtract fractions, first rewrite each fraction as an equivalent fraction, all having the same denominator, then add or subtract the numerators and place the result over the common denominator.
  • [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]
• To multiply fractions, place the product of the numerators over the product of the denominator.
  • [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
• To divide fractions, multiply the first fraction by the reciprocal of the second
  • [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex].
• To simplify a fraction, rewrite the numerator and denominator as products of their prime factors, then cancel ratios of common factors until there are no more common factors between the top and the bottom.
• Division by zero is undefined.
• When applying the order of operations, first simplify inside grouping symbols, then evaluate exponents or radicals, then multiply or divide from left to right in the order each appears, and finally add or subtract from left to right in the order each appears.