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MathChapter 1: The Language and Tools of Algebra

Notes

Exponents and Order of Operations Notes

Interpret powers and evaluate nested expressions with correct grouping and left-to-right priority rules.

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Exponents compress repeated multiplication, and order of operations makes an expression unambiguous. SAT distractors often encode one plausible wrong order, so a written sequence is safer than mental shortcuts.

Base, exponent, and power

Base
The repeated factor. In \(5^3\), the base is \(5\).
Exponent
The number of times the base is used as a factor. In \(5^3\), the exponent is \(3\).
Power
An expression consisting of a base and exponent, or its value: \(5^3=5\cdot5\cdot5=125\).

The exact order of operations

A reliable evaluation sequence

  1. Grouping symbols

    Work from the innermost parentheses, brackets, braces, absolute values, fraction bars, or radical groupings outward.

  2. Exponents

    Evaluate powers after their bases and exponents are known.

  3. Multiplication and division

    They have equal priority. Perform them from left to right.

  4. Addition and subtraction

    They also have equal priority. Perform them from left to right.

Worked example

Evaluate a nested expression

Evaluate \(100-2[3^2+4(5-3)]\).

  1. Interpret

    Inside the bracket, evaluate \(5-3=2\) and the power \(3^2=9\).

  2. Set up

    Complete multiplication: \(4(2)=8\), so the bracket is \(9+8=17\).

  3. Work

    Multiply outside the bracket: \(2(17)=34\).

  4. Check

    Finish the subtraction: \(100-34=66\).

The expression equals \(66\).

Substitution and evaluation

Substitution replaces a variable with a value; it does not change the surrounding operations. If \(x=-3\), then \(-x^2+2x=-(-3)^2+2(-3)=-9-6=-15\). Parentheses make the substituted sign visible.

Mini check

Check your understanding

Evaluate \(18\div3\times2+2^3\).

  1. \(11\)
  2. \(20\)
  3. \(32\)
  4. \(14\)
Show answer and explanation

Answer: \(20\)

Evaluate \(2^3=8\). Then multiplication/division left to right gives \(18\div3\times2=12\), and \(12+8=20\).

Key takeaways

Key takeaways

What to remember

  • The base is the complete grouped quantity to which an exponent applies.
  • Grouping precedes exponents; multiplication/division tie; addition/subtraction tie.
  • At equal priority, work left to right.
  • Parenthesize negative substitutions and keep one calculation line per priority level.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.