ORIGINAL WORKED EXAMPLE
Simplify before calculating
Evaluate (3⁵·3⁻²)/3².
- Combine the numerator exponents: 3^(5−2)=3³.
- Subtract the denominator exponent: 3^(3−2)=3.
- All bases are nonzero, so the laws apply.
Answer 3
GRE · Arithmetic · TOPIC LESSON
Powers, roots and numerical patterns
Track bases, signs and domains before simplifying powers, radicals and numerical patterns.
BY THE END, YOU CAN
- Apply exponent laws with their restrictions.
- Separate principal roots from equation solutions.
- Analyze recursive, arithmetic, geometric and repeating sequences.
BEFORE YOU START
- Signed multiplication
- Fractions and prime factorization
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
Original GRE-style worked examples and practice
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
An exponent describes repeated multiplication
For positive integer n, aⁿ is a product of n copies of a. A negative base raised to an even integer is nonnegative; an odd integer power preserves the negative sign. Parentheses matter: (−4)²=16, while −4²=−16. An exponent does not mean repeated addition.
For nonzero a, a⁰=1 and a⁻ⁿ=1/aⁿ. A negative exponent means reciprocal, not a negative value. Zero to a negative exponent is undefined because it would require division by zero; do not apply the zero-exponent rule to 0⁰.
02 / UNDERSTAND & APPLY
Know which operations combine exponents
For the same nonzero base, aᵐaⁿ=aᵐ⁺ⁿ and aᵐ/aⁿ=aᵐ⁻ⁿ. A power of a power multiplies integer exponents: (aᵐ)ⁿ=aᵐⁿ wherever defined. A power of a product distributes to the factors; a power does not distribute over addition. For example, (2+3)²=25, not 4+9.
When bases differ, rewrite them as powers of a common base if possible. Otherwise calculate or factor rather than inventing a law. Compare 2¹⁰ and 4⁵ by rewriting 4 as 2²; they are equal.
03 / UNDERSTAND & APPLY
The radical symbol chooses one root
For a≥0, √a is the nonnegative number whose square is a. Thus √49=7, but the equation x²=49 has solutions x=7 and x=−7. For any real x, √(x²)=|x|, not necessarily x. Odd roots retain signs: the real cube root of −27 is −3.
For nonnegative a and b, √(ab)=√a√b. Pull out perfect-square factors to simplify radicals. A sum under a radical cannot be split: √(9+16)=5, whereas √9+√16=7. Even roots of negative numbers are outside real-number arithmetic.
ORIGINAL WORKED EXAMPLE
Simplify with a sign condition
If x<0, simplify √(50x²).
- Factor 50x²=25·2·x².
- The radical equals 5√2·|x|.
- Because x<0, |x|=−x, so the expression is −5x√2, a positive quantity.
Answer −5x√2
04 / UNDERSTAND & APPLY
Powers and roots can reverse naive size expectations
For a fixed base a>1, aⁿ increases as n increases. Do not apply this rule when both the base and exponent change. More usefully, for fixed x>1, x²>x, while for 0<x<1, x²<x. For x<0, signs and exponent parity must also be considered.
To compare nonnegative quantities, squaring preserves order. It does not preserve order across arbitrary negative numbers: −5<−2 but 25>4. For root estimates, bracket the radicand between perfect squares. If 64<70<81, then 8<√70<9.
ORIGINAL WORKED EXAMPLE
QC demands more than an integer trial
QC: x>0. Quantity A: x². Quantity B: x.
- At x=2, A=4>B=2.
- At x=1/2, A=1/4<B=1/2.
- Both inputs are permitted; different relationships establish D.
Answer D
05 / UNDERSTAND & APPLY
Separate the coefficient from the power of ten
Scientific notation writes a nonzero magnitude as c×10ⁿ with 1≤c<10 and integer n; a negative number carries a negative sign. Multiplication combines coefficients and adds exponents; division divides coefficients and subtracts exponents. Renormalize if the coefficient is outside the interval.
When adding quantities, first express them with the same power of ten. Counting decimal shifts mechanically without a size estimate invites errors. A tiny number multiplied by a large one need not be tiny.
ORIGINAL WORKED EXAMPLE
A product across scales
Calculate (6×10⁻⁴)(5×10⁷) in scientific notation.
- Multiply coefficients to get 30. Add exponents: −4+7=3.
- 30×10³=3×10⁴ after normalization.
- The ordinary value is 30000.
Answer 3×10⁴
06 / UNDERSTAND & APPLY
An explicit rule and a recurrence answer different questions
An arithmetic sequence adds a fixed difference d: aₙ=a₁+(n−1)d. A geometric sequence multiplies by a fixed ratio r: aₙ=a₁rⁿ⁻¹. Indexing starts at the specified first term; using n in place of n−1 shifts the sequence. A recurrence such as aₙ₊₁=2aₙ−1 tells how to move from one term to the next.
A short list of terms alone need not determine a unique rule. Use the rule the problem states. For an arithmetic sum, pair the first and last terms: the average term is (first+last)/2, so the sum is that average times the number of terms.
ORIGINAL WORKED EXAMPLE
Arithmetic position and total
A sequence starts at 7 and increases by 4. Find the 12th term and sum of the first 12 terms.
- a₁₂=7+11·4=51.
- The average of the first and last term is (7+51)/2=29.
- Twelve terms sum to 12·29=348.
Answer 51 and 348
07 / UNDERSTAND & APPLY
Use remainders to locate repeating patterns
A pattern of length k repeats every k positions. Divide the requested index by k; a remainder of 1 refers to the first entry, and remainder 0 to the last entry, if indexing starts at 1. Last-digit patterns of integer powers are a common application.
For powers of 7, the last digits cycle 7,9,3,1. Exponent 20 has remainder 0 on division by 4, so 7²⁰ ends in 1. Do not mistake zero remainder for a nonexistent zeroth item.
08 / UNDERSTAND & APPLY
Choose a method and answer the requested question
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
09 / UNDERSTAND & APPLY
Original practice: laws, roots and patterns
Use exact values unless a decimal is explicitly requested.
ORIGINAL PRACTICE · 1
Numeric entry: Evaluate 2⁻³+2⁻².
Reveal answer & reasoning
3/8
The terms are 1/8 and 1/4, which sum to 3/8. Negative exponents do not make the terms negative.
ORIGINAL PRACTICE · 2
Single answer: For real x<0, √(x²) equals A x; B −x; C x²; D −x²; E 0.
Reveal answer & reasoning
B
A principal square root is nonnegative. Since x<0, |x|=−x. Choosing x would give a negative square root.
ORIGINAL PRACTICE · 3
Select all that apply: Which real x satisfy x²=81? A −9; B 9; C 81; D −81; E 3.
Reveal answer & reasoning
A and B
Both ±9 square to 81. The expression √81 equals only 9, but this is an equation asking for every solution.
ORIGINAL PRACTICE · 4
QC: 0<t<1. Quantity A: √t. Quantity B: t.
Reveal answer & reasoning
A
For 0<t<1, t²<t. Both t and √t are positive; equivalently divide √t by t to get 1/√t>1. Testing only t=1 is invalid because that endpoint is excluded.
ORIGINAL PRACTICE · 5
Numeric entry: A geometric sequence has first term 3 and ratio −2. What is its fifth term?
Reveal answer & reasoning
48
The fifth term is 3(−2)⁴=48. There are four ratio multiplications after the first term, not five.
ORIGINAL PRACTICE · 6
Single answer: Which bounds contain √115? A 9 and 10; B 10 and 11; C 11 and 12; D 12 and 13; E 13 and 14.
Reveal answer & reasoning
B
100<115<121, so 10<√115<11. Comparing to squares, rather than halving 115, provides reliable bounds.
ORIGINAL PRACTICE · 7
Numeric entry: What is the units digit of 7²³?
Reveal answer & reasoning
3
The cycle 7,9,3,1 has length four. 23 leaves remainder 3, so use the third entry. Do not use the exponent itself as the digit.
ORIGINAL PRACTICE · 8
Numeric entry: A recurrence has a₁=4 and aₙ₊₁=3aₙ−2. Find a₃.
Reveal answer & reasoning
28
a₂=3·4−2=10; a₃=3·10−2=28. The rule is not multiplication by a constant ratio, so a geometric formula cannot be used directly.
KEEP THE BIG IDEAS
Quick revision
- Negative exponents mean reciprocals; they require nonzero bases.
- √(x²)=|x|, while x²=k may have two real solutions.
- Check 0<x<1 before assuming powers grow.
- Respect starting indices and cycle remainder zero.
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