MTH1W · Unit 1 test review

Number Sets and Powers: Test Review

Everything for the Unit 1 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Number Sets: Natural to Real

$N \subset W \subset Z \subset Q \subset R$ ; the irrational numbers are real but sit outside $Q$ .
Rational: can be written as $\frac{a}{b}$ with integers $a$ , $b$ and $b \neq = 0$ . Its decimal terminates or repeats.
Irrational: its decimal never ends and never repeats (e.g. $2$ , $π$ ).
For a whole number $n$ , $n$ is rational only when $n$ is a perfect square.
Density: between any two different rational numbers there is another, such as their average $\frac{a + b}{2}$ .
The running totals of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ approach the limit 1 without ever passing it; the endless sum equals 1.

Numbers Across Cultures and Time

In base $b$ , the places are worth $1, b, b^{2}, b^{3}, \dots$ ; e.g. base 60 places are 1, 60, 3600.
Babylonian base 60 survives in 60 seconds per minute and 60 minutes per hour; their zero was a placeholder sign (by about 300 BCE).
Maya numerals: dot = 1, bar = 5, shell = 0, stacked with ones at the bottom and twenties above (base 20).
Brahmagupta (628 CE) gave rules for zero and negative numbers as numbers; Hindu–Arabic base 10 digits were popularized in Europe by Fibonacci (1202).
Chinese counting rods (Nine Chapters, Liu Hui 263 CE): red rods positive, black rods negative.
Golden ratio $φ = \frac{1 + 5}{2} \approx 1.618$ , irrational; ratios of neighbouring Fibonacci numbers approach it.

Powers, Zero and Negative Exponents, Scientific Notation

$a^{n}$ means $n$ factors of $a$ multiplied together (for a natural number $n$ ); $a^{1} = a$ .
$(- 2)^{4} = 16$ but $- 2^{4} = - 16$ : without brackets, the exponent applies before the negative sign.
$a^{0} = 1$ for any $a \neq = 0$ .
$a^{- n} = \frac{1}{a ^{n}}$ for $a \neq = 0$ ; a negative exponent gives a reciprocal, not a negative number ( $2^{- 3} = \frac{1}{8}$ ).
Scientific notation: $a \times 1 0^{n}$ with $1 \leq a < 10$ and $n$ an integer; positive $n$ for large numbers, negative $n$ for numbers between 0 and 1.

The Exponent Laws

Product law: $a^{m} \times a^{n} = a^{m + n}$ (same base: keep it, add exponents).
Quotient law: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ for $a \neq = 0$ .
Power of a power: $(a^{m})^{n} = a^{mn}$ .
Power of a product: $(ab)^{n} = a^{n} b^{n}$ ; e.g. $(2 x)^{3} = 8 x^{3}$ .
The laws apply to multiplying and dividing powers, never to adding them: $x^{2} + x^{3} \neq = x^{5}$ .

Where marks get lost

Thinking every square root is irrational. $49 = 7$ is a natural number; only square roots of non-perfect squares (like $50$ ) are irrational. (Number Sets: Natural to Real)
Treating $π$ as rational because " $π = 3.14$ " or " $π = \frac{22}{7}$ ." Both are only approximations; $π$ itself is irrational. (Number Sets: Natural to Real)
Putting a number in only one set. The number 5 is natural and whole and an integer and rational and real. (Number Sets: Natural to Real)
Calling 0 a natural number in this course, or calling $- 3$ a whole number. Natural numbers start at 1; whole numbers include 0 but no negatives. (Number Sets: Natural to Real)
Deciding a decimal terminates because the calculator screen stops. The screen rounds to about 10 digits; $3$ looks like 1.732050808 but continues forever. (Number Sets: Natural to Real)
Reading a Maya number from the top as the ones place. The bottom level is the ones; each level up is worth 20 times more. (Numbers Across Cultures and Time)
Thinking base 60 needs 60 different symbols. The Babylonians built every digit from 1 to 59 out of just two wedge shapes. (Numbers Across Cultures and Time)
Treating a placeholder zero as the same thing as the number zero. A placeholder only marks an empty place; using zero in calculations came later. (Numbers Across Cultures and Time)
Assuming red always means negative. On Chinese counting boards, red rods were positive and black rods were negative. (Numbers Across Cultures and Time)
Writing $φ = 1.618$ exactly. The golden ratio is irrational; 1.618 is a rounded value. (Numbers Across Cultures and Time)
Multiplying the base by the exponent: writing $2^{3} = 6$ instead of $2 \times 2 \times 2 = 8$ . (Powers, Zero and Negative Exponents, Scientific Notation)
Treating $- 2^{4}$ as $(- 2)^{4} = 16$ . Without brackets the exponent applies first, so $- 2^{4} = - 16$ . (Powers, Zero and Negative Exponents, Scientific Notation)
Thinking a negative exponent makes the answer negative: $2^{- 3}$ is $\frac{1}{8}$ , not $- 8$ . (Powers, Zero and Negative Exponents, Scientific Notation)
Saying $5^{0} = 0$ or $5^{0} = 5$ . Any nonzero base to the exponent 0 equals 1. (Powers, Zero and Negative Exponents, Scientific Notation)
Leaving the coefficient outside the range 1 to 10, as in $45 \times 1 0^{6}$ (should be $4.5 \times 1 0^{7}$ ), or giving a small number a positive exponent. (Powers, Zero and Negative Exponents, Scientific Notation)
Multiplying the bases: writing $2^{3} \times 2^{4} = 4^{7}$ . The base stays 2, so the answer is $2^{7}$ . (The Exponent Laws)
Multiplying exponents in a product: writing $x^{3} \cdot x^{4} = x^{12}$ . Multiply exponents only for a power of a power, $(x^{3})^{4} = x^{12}$ ; here add them, $x^{7}$ . (The Exponent Laws)
Combining different bases: writing $2^{3} \times 3^{2} = 6^{5}$ . There is no shortcut; evaluate $8 \times 9 = 72$ . (The Exponent Laws)
Forgetting to apply the exponent to the coefficient: writing $(2 x)^{3} = 2 x^{3}$ instead of $8 x^{3}$ . (The Exponent Laws)
Adding exponents when adding terms: writing $x^{2} + x^{3} = x^{5}$ . The laws only work for multiplying and dividing. (The Exponent Laws)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Number Sets: Natural to Real
1
Which number is irrational?
Number Sets: Natural to Real
2
Order the number sets from the smallest (innermost) to the largest (outermost).
1. 1Real numbers
2. 2Rational numbers
3. 3Integers
4. 4Natural numbers
5. 5Whole numbers
Number Sets: Natural to Real
3
Find the number exactly halfway between $\frac{1}{4}$ and $\frac{1}{2}$ . Give it as a fraction or an exact decimal.
Number Sets: Natural to Real
4
Which statement is true?
Number Sets: Natural to Real
5
Write $0. \overline{7} = 0.777 \dots$ as a fraction.
Number Sets: Natural to Real
6
As you keep adding terms of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ , what happens to the running total?
Numbers Across Cultures and Time
7
A Maya number shows one dot on the top level and a shell on the bottom level. What is its value in base 10?
Numbers Across Cultures and Time
8
Who, in 628 CE, wrote rules for calculating with zero and with negative numbers as numbers?
Numbers Across Cultures and Time
9
Why is an hour split into 60 minutes?
Numbers Across Cultures and Time
10
What symbol did the Maya use to write zero?
Numbers Across Cultures and Time
11
Compute the Fibonacci ratio $\frac{21}{13}$ to three decimal places.
Numbers Across Cultures and Time
12
How many whole numbers from 1 to 60 divide 60 with no remainder?
Powers, Zero and Negative Exponents, Scientific Notation
13
Evaluate $3^{4}$ .
Powers, Zero and Negative Exponents, Scientific Notation
14
Evaluate $(- 2)^{5}$ .
Powers, Zero and Negative Exponents, Scientific Notation
15
What is the value of $7^{0}$ ?
Powers, Zero and Negative Exponents, Scientific Notation
16
Order these values from smallest to largest.
1. 1 $3^{2}$
2. 2 $2^{3}$
3. 3 $5^{0}$
4. 4 $1 0^{- 3}$
5. 5 $2^{- 1}$
Powers, Zero and Negative Exponents, Scientific Notation
17
0.0007 is written as $7 \times 1 0^{n}$ . What is $n$ ?
Powers, Zero and Negative Exponents, Scientific Notation
18
A sheet of paper 0.1 mm thick is folded in half 8 times, doubling its thickness each time. How thick is the folded stack, in millimetres?
mm
The Exponent Laws
19
Simplify $x^{4} \cdot x^{6}$ .
The Exponent Laws
20
Simplify $(3 y^{2})^{3}$ .
The Exponent Laws
21
Simplify $x^{5} \cdot x \cdot x^{2}$ . Type powers with ^: type x^3 for $x^{3}$ .
The Exponent Laws
22
Evaluate $\frac{( 1 0 ^{3} ) ^{2} \times 1 0 ^{4}}{1 0 ^{7}}$ .
The Exponent Laws
23
Put the steps for simplifying $\frac{( 2 x ^{3} ) ^{2} \cdot x ^{4}}{4 x ^{5}}$ in order.
1. 1Multiply by $x^{4}$ with the product law: $4 x^{6} \cdot x^{4} = 4 x^{10}$
2. 2Write the final answer: $x^{5}$
3. 3Divide by $4 x^{5}$ : $4 \div 4 = 1$ and $x^{10 - 5} = x^{5}$
4. 4Apply the power to the brackets: $(2 x^{3})^{2} = 4 x^{6}$
The Exponent Laws
24
Multiply $(2 \times 1 0^{3}) (4 \times 1 0^{5})$ .

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…