Integers, Exponents, Prime Time, Lines & Angles, Kinematics, and Matter: 33 flashcards
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1. What is an integer?
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A whole number that can be positive, negative, or zero: $\ldots, -2, -1, 0, 1, 2, \ldots$
2. In $a^n$, what are $a$ and $n$ called?
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$a$ is the base and $n$ is the exponent (or index).
3. What does $a^n$ mean for a positive integer $n$?
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$a$ multiplied by itself $n$ times: $a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$.
4. State the product rule for exponents.
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$a^m \times a^n = a^{m+n}$ for the same base $a$.
5. State the quotient rule for exponents.
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$\dfrac{a^m}{a^n} = a^{m-n}$ for $m > n$ and the same base $a$.
6. State the power-of-a-power rule.
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$(a^m)^n = a^{mn}$ — a power raised to a power multiplies the exponents.
7. State the power-of-a-product rule.
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$(ab)^n = a^n b^n$ — a power distributes over a product.
8. What is $a^0$ for any non-zero $a$?
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$a^0 = 1$ for any non-zero $a$.
9. What does a negative exponent mean?
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$a^{-n} = \dfrac{1}{a^n}$ — a negative exponent means the reciprocal of the positive power.
10. Evaluate $(-2)^4$ and $-2^4$.
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$(-2)^4 = 16$ but $-2^4 = -16$; brackets change the base.
11. State the power-of-a-quotient rule.
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$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ for $b \neq 0$.
12. What is a prime number?
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An integer greater than 1 with exactly two positive divisors: 1 and itself.
13. How can prime factorisation be written compactly?
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Using exponents, e.g. $360 = 2^3 \times 3^2 \times 5$.
14. What is a pattern in mathematics?
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A sequence governed by a rule; recognising the rule lets you predict later terms.
15. What is an angle and what is its vertex?
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An angle is formed by two rays meeting at a common endpoint called the vertex.
16. Classify angles by size.
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Acute ($<90^\circ$), right ($=90^\circ$), obtuse (between $90^\circ$ and $180^\circ$), straight ($=180^\circ$), reflex (between $180^\circ$ and $360^\circ$).
17. What do angles on a straight line and around a point add up to?
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Angles on a straight line add to $180^\circ$; angles around a point add to $360^\circ$.
18. What is kinematics?
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The description of motion using distance, displacement, speed, velocity, and acceleration.
19. Distinguish distance and displacement.
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Distance is a scalar (magnitude only); displacement is a vector with direction.
20. Define speed and velocity.
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Speed $= \dfrac{\text{distance}}{\text{time}}$; velocity is speed with direction.
21. Define acceleration.
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Acceleration is the rate of change of velocity with time.
22. What are the three common states of matter?
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Solid, liquid, and gas.
23. What are two key properties of particles of matter?
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They are in constant motion and have spaces between them.
24. What can change the state of matter?
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Changing temperature or pressure can change the state of matter.
25. Why are exponents useful in science?
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They write very large or very small quantities compactly, e.g. $3 \times 10^8\ \text{m/s}$ or $10^{-9}\ \text{m}$.
26. Why does $a^m \times a^n = a^{m+n}$?
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Multiplying two powers with the same base strings the multiplications together: $2^3 \times 2^4$ is $(2 \times 2 \times 2)(2 \times 2 \times 2 \times 2)$, i.e. seven 2s multiplied, so $2^7$. The exponents add because the count of factors adds.
27. Why does $\dfrac{a^m}{a^n} = a^{m-n}$ (for $m > n$)?
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The shared factors in numerator and denominator cancel, leaving $m - n$ factors of $a$. Example: $\dfrac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$.
28. Why is $(a^m)^n = a^{mn}$?
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The inner power $a^m$ is multiplied by itself $n$ times, so the total count of $a$'s is $m \times n$. Example: $(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$.
29. How do you decide which exponent rule applies to an expression?
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Look at the operation on the outside: multiplying powers with the same base → add exponents; dividing → subtract exponents; a power raised to a power → multiply exponents; a product raised to a power → distribute the exponent over each factor.
30. Why must $a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$?
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They are forced by the division rule, not invented. Since $\dfrac{a^3}{a^3} = 1$ and also equals $a^{3-3} = a^0$, we get $a^0 = 1$. Since $\dfrac{a^2}{a^5} = a^{-3}$, a negative exponent means a reciprocal: $a^{-n} = \dfrac{1}{a^n}$.
31. Simplify $\dfrac{(2^3)^2 \times 2^4}{2^5}$ step by step.
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Step 1: $(2^3)^2 = 2^{3 \times 2} = 2^6$. Step 2: $2^6 \times 2^4 = 2^{6+4} = 2^{10}$. Step 3: $\dfrac{2^{10}}{2^5} = 2^{10-5} = 2^5 = 32$. Each step uses exactly one rule.
32. How are exponents used in kinematics and in describing matter?
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Powers of ten write very large or very small quantities compactly: speeds like $3 \times 10^8\ \text{m/s}$ in kinematics, and particle sizes like $10^{-9}\ \text{m}$ in chemistry. Measurements such as temperature, mass, and volume are also expressed with numbers, sometimes with exponents.
33. Is an exponent a new kind of number?
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No — an exponent is only a compact way to write repeated multiplication. $2^5$ is just 32 written differently. Zero and negative exponents are not new inventions either; they are what the rules demand so that $a^m \div a^n = a^{m-n}$ works for every pair of integers $m, n$.
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