Math Operations and Simplifications Summary

Order of Operations

  • Start with parentheses, then addition and subtraction left to right.

  • Remember PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Exponents and Simplification

  • Any number to the power of one equals itself (e.g., 51=55^1 = 5).

  • Any number to the power of zero equals one (e.g., 90=19^0 = 1).

  • For multiplication with like bases, add exponents (e.g., 32×34=32+4=363^2 \times 3^4 = 3^{2+4} = 3^6).

  • For division with like bases, subtract exponents (e.g., 106÷102=106−2=10410^6 \div 10^2 = 10^{6-2} = 10^4).

Fraction Operations

  • Addition/Subtraction: Find common denominators.

    • e.g., 56−13→56−26=36=12\frac{5}{6} - \frac{1}{3} \rightarrow \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}.

  • Multiplication: Multiply straight across (e.g., 35×23=615→25\frac{3}{5} \times \frac{2}{3} = \frac{6}{15} \rightarrow \frac{2}{5}).

  • Division: Multiply by the reciprocal (e.g., 34÷12=34×21=64=32\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}).

Converting Decimals to Fractions

  • Identify the place value (0.25 = 25100=14\frac{25}{100} = \frac{1}{4}).

  • For 0.125, move decimal three places for thousandths: 1251000=18\frac{125}{1000} = \frac{1}{8}.

Decimal to Percentage

  • Convert by multiplying the decimal by 100 (e.g., 0.45 = 45%).

  • For U.S. problems use: x% of y = (x/100)*y.