Seed content: Quant arithmetic exponents & roots (lesson + 6 questions)

content/lessons/quant/quant-arithmetic-exponents-roots.md

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+---
+id: quant-arithmetic-exponents-roots
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+title: "Exponents & Roots"
+tags: [exponents, roots, powers, radicals]
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Overview
+
+Exponents and roots show up all over GMAT Focus Quant — in number properties, algebra, and word
+problems (growth, interest, doubling). Almost every question is solved by applying a small set of
+**exponent rules** rather than by grinding out large powers. Learn the rules cold and most of
+these problems become one or two lines.
+
+## Core concepts
+
+**The exponent rules.** For any nonzero base and integer exponents:
+
+\[a^m \cdot a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}\]
+
+\[(ab)^n = a^n b^n, \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\]
+
+**Zero and negative exponents.**
+
+\[a^0 = 1 \ (a \neq 0), \qquad a^{-n} = \frac{1}{a^n}\]
+
+A negative exponent means "reciprocal," not "negative number": \(2^{-3} = \tfrac{1}{8}\), not \(-8\).
+
+**Common-base is the master move.** To compare or solve, rewrite everything as a power of the same
+base. If \(a^x = a^y\) (and \(a \neq 0, \pm 1\)), then \(x = y\). Example: \(4^3 = (2^2)^3 = 2^6\),
+so \(4^3\) and \(8^2 = (2^3)^2 = 2^6\) are equal.
+
+**Roots are fractional exponents.**
+
+\[\sqrt{a} = a^{1/2}, \qquad \sqrt[n]{a} = a^{1/n}, \qquad \sqrt[n]{a^m} = a^{m/n}\]
+
+So the exponent rules apply to roots too. Simplify radicals by pulling out perfect-square factors:
+\(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\).
+
+**Multiply and divide radicals** freely: \(\sqrt{a}\,\sqrt{b} = \sqrt{ab}\) and
+\(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\tfrac{a}{b}}\). But **you cannot add** across a root:
+\(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\).
+
+## Worked examples
+
+**Common base.** Solve \(2^{x} = 32\). Since \(32 = 2^5\), \(x = 5\).
+
+**Combine, don't compute.** \(\dfrac{3^{10}}{3^{7}} = 3^{10-7} = 3^3 = 27\) — never expand \(3^{10}\).
+
+**Factor out the common power.** \(2^{11} + 2^{11} = 2 \cdot 2^{11} = 2^{12}\). Adding equal powers
+*doubles*; it does not add the exponents.
+
+**Simplify a radical.** \(\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}\).
+
+## Common traps
+
+- **Adding exponents when multiplying bases.** \(2^3 \cdot 3^3 \neq 6^6\). Same *exponent*, different
+  base: \(2^3 \cdot 3^3 = (2\cdot 3)^3 = 6^3\). Add exponents only for the *same base*.
+- **Adding across a root or a sum of powers.** \(\sqrt{9+16} = \sqrt{25} = 5\), not \(3+4=7\).
+- **Mishandling negative exponents.** \(a^{-n}\) is a reciprocal, not a negative value.
+- **Forgetting \(a^0 = 1\).** Any nonzero base to the zero power is 1, regardless of the base.
+
+## Key takeaways
+
+- Multiply → add exponents; divide → subtract; power of a power → multiply. Same base only.
+- Rewrite everything to a **common base** to compare or solve equations with exponents.
+- Roots are fractional exponents; simplify by extracting perfect-square factors.
+- You can multiply and divide radicals, but you cannot split a root over addition.

content/questions/quant/quant-arithmetic-exponents-roots-0017.md

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+---
+id: quant-arithmetic-exponents-roots-0017
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: easy
+tags: [exponents, common-base]
+choices:
+  A: "3"
+  B: "4"
+  C: "5"
+  D: "6"
+  E: "8"
+answer: C
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+If \(2^{x} = 32\), what is the value of \(x\)?
+
+## Explanation
+
+Rewrite the right side as a power of the same base, 2. Since \(32 = 2 \times 2 \times 2 \times 2
+\times 2 = 2^5\), the equation becomes:
+
+\[2^{x} = 2^{5}\]
+
+When two equal powers share the same base, their exponents must be equal, so \(x = 5\).
+
+**The trap:** answer **E (8)** comes from computing \(32 \div 4\) or otherwise treating the problem
+as arithmetic rather than matching exponents. Don't multiply — match the base.
+
+## Hints
+
+- Write 32 as a power of 2.
+- If \(2^x = 2^5\), what must \(x\) be?

content/questions/quant/quant-arithmetic-exponents-roots-0018.md

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+---
+id: quant-arithmetic-exponents-roots-0018
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: easy
+tags: [exponents, division, common-base]
+choices:
+  A: "9"
+  B: "27"
+  C: "81"
+  D: "243"
+  E: "729"
+answer: B
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+What is the value of \(\dfrac{3^{10}}{3^{7}}\)?
+
+## Explanation
+
+Never expand \(3^{10}\). When you divide powers of the same base, **subtract** the exponents:
+
+\[\frac{3^{10}}{3^{7}} = 3^{10 - 7} = 3^{3}\]
+
+And \(3^3 = 27\).
+
+**The trap:** answer **E (729)** is \(3^6\), which you'd get by mistakenly *dividing* the exponents
+(\(10 \div 7\) rounded, or similar) or by other slips. Division of like bases means subtracting
+exponents, not dividing them.
+
+## Hints
+
+- Same base, division → subtract the exponents.
+- Compute \(3^{10-7}\).

content/questions/quant/quant-arithmetic-exponents-roots-0019.md

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+---
+id: quant-arithmetic-exponents-roots-0019
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: medium
+tags: [exponents, negative-exponents, fractions]
+choices:
+  A: "-8"
+  B: "-\\tfrac{1}{8}"
+  C: "\\tfrac{1}{8}"
+  D: "\\tfrac{1}{6}"
+  E: "8"
+answer: C
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+What is the value of \(2^{-3}\)?
+
+## Explanation
+
+A negative exponent means "take the reciprocal," not "make the result negative." By the rule
+\(a^{-n} = \dfrac{1}{a^{n}}\):
+
+\[2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}\]
+
+**The trap:** answer **A (\(-8\))** treats the negative sign as if it made the value negative, and
+**B** repeats that error inside the fraction. The exponent's sign controls reciprocal-vs-not, not
+the sign of the answer. Choice **D (\(\tfrac{1}{6}\))** comes from computing \(2 \times 3\) in the
+denominator instead of \(2^3\).
+
+## Hints
+
+- \(a^{-n}\) is the reciprocal of \(a^{n}\), and stays positive.
+- The denominator is \(2^3\), not \(2 \times 3\).

content/questions/quant/quant-arithmetic-exponents-roots-0020.md

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+---
+id: quant-arithmetic-exponents-roots-0020
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: medium
+tags: [exponents, adding-powers, factoring]
+choices:
+  A: "2^{11}"
+  B: "2^{12}"
+  C: "2^{22}"
+  D: "4^{11}"
+  E: "4^{22}"
+answer: B
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+Which of the following is equal to \(2^{11} + 2^{11}\)?
+
+## Explanation
+
+You are **adding** two equal quantities, so the result is twice one of them:
+
+\[2^{11} + 2^{11} = 2 \times 2^{11}\]
+
+Now \(2 = 2^1\), and multiplying like bases means adding exponents:
+
+\[2^{1} \times 2^{11} = 2^{1+11} = 2^{12}\]
+
+**The trap:** answer **C (\(2^{22}\))** comes from adding the exponents (\(11 + 11\)) as if the two
+terms were multiplied. They are added, so the powers double — increasing the exponent by exactly 1.
+Choice **D (\(4^{11}\))** looks tempting but \(4^{11} = 2^{22}\), the same wrong value.
+
+## Hints
+
+- Adding two equal terms means multiplying one of them by 2.
+- Write the leading 2 as \(2^1\), then combine like bases.

content/questions/quant/quant-arithmetic-exponents-roots-0021.md

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+---
+id: quant-arithmetic-exponents-roots-0021
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: medium
+tags: [roots, radicals, simplifying]
+choices:
+  A: "5\\sqrt{2}"
+  B: "8\\sqrt{2}"
+  C: "\\sqrt{68}"
+  D: "34"
+  E: "68"
+answer: B
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+What is \(\sqrt{50} + \sqrt{18}\) in simplest form?
+
+## Explanation
+
+Simplify each radical by pulling out its largest perfect-square factor, then combine like terms.
+
+\[\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\]
+\[\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\]
+
+Both are multiples of \(\sqrt{2}\), so add the coefficients:
+
+\[5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}\]
+
+**The trap:** answer **C (\(\sqrt{68}\))** comes from adding *under* the radical
+(\(\sqrt{50+18}\)) — you cannot split or combine a root over addition that way. You must simplify
+each radical first, then add the like radical terms.
+
+## Hints
+
+- \(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\) — don't add inside the root.
+- Extract perfect-square factors from 50 and 18, then combine the \(\sqrt{2}\) terms.

content/questions/quant/quant-arithmetic-exponents-roots-0022.md

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+---
+id: quant-arithmetic-exponents-roots-0022
+section: quant
+topic: arithmetic
+subtopic: exponents-roots
+type: problem-solving
+difficulty: hard
+tags: [exponents, common-base, power-of-a-power]
+choices:
+  A: "2"
+  B: "3"
+  C: "4"
+  D: "5"
+  E: "6"
+answer: A
+author: openmat
+reviewers: []
+status: in-review
+original: true
+license: CC-BY-SA-4.0
+---
+
+## Question
+
+If \(9^{x} \cdot 27 = 3^{7}\), what is the value of \(x\)?
+
+## Explanation
+
+Rewrite every term as a power of the common base 3.
+
+- \(9 = 3^2\), so \(9^{x} = (3^2)^{x} = 3^{2x}\) (power of a power → multiply exponents).
+- \(27 = 3^3\).
+
+Substitute:
+
+\[3^{2x} \cdot 3^{3} = 3^{7}\]
+
+Multiplying like bases adds exponents on the left:
+
+\[3^{2x + 3} = 3^{7}\]
+
+Now the bases match, so set the exponents equal:
+
+\[2x + 3 = 7 \;\Rightarrow\; 2x = 4 \;\Rightarrow\; x = 2\]
+
+**The trap:** answer **C (4)** is the value of \(2x\) — easy to stop one step early. Answer **D (5)**
+comes from forgetting to rewrite 27 and using \(2x + 2 = 7\) or a similar slip. Convert *every* term
+to base 3 before equating exponents.
+
+## Hints
+
+- Rewrite 9 and 27 as powers of 3.
+- \((3^2)^x = 3^{2x}\); then add exponents on the left and set them equal to 7.

curriculum.md

@@ -70,7 +70,7 @@ The taxonomy is ordered so you can study a section top to bottom. A balanced pla
 Early days — the bank is growing. Rough coverage so far (a contributor could automate this
 count from the front-matter):
 
-- **Quant** — arithmetic, number properties, algebra (linear, inequalities), word problems (rates, work, sets), counting & probability, statistics (mean, median, weighted average, standard deviation).
+- **Quant** — arithmetic (fractions, percents/ratios, exponents & roots), number properties, algebra (linear, quadratics, inequalities), word problems (rates, work, sets), counting & probability, statistics (mean, median, weighted average, standard deviation).
 - **Verbal** — Critical Reasoning (assumptions, strengthen/weaken, inference, flaw) and Reading Comprehension (main idea, detail/inference).
 - **Data Insights** — Data Sufficiency across fundamentals, divisibility, number properties, algebra, statistics, word problems, and overlapping sets.