Real numbers

Advanced Math 7 · Unit 2 · reals-radicals-sci-notation · Teacher edition

reals-radicals-sci-notation

Name ______________________________Date ______________

1.
$\sqrt{50}$ lies between two consecutive whole numbers. Enter the smaller one.
Answer: ______________
2.
Write $4{,}560{,}000$ in scientific notation $a \times 10^k$ with $1 \le a < 10$ . Enter $a$ .
Answer: ______________
3.
Which list contains only rational numbers?
1. (A)
  $\sqrt{50},\ \dfrac{22}{7},\ 0.5$
2. (B)
  $\pi,\ 3.14,\ \dfrac{22}{7}$
3. (C)
  $\sqrt{2},\ \sqrt{4},\ \sqrt{8}$
4. (D)
  $\sqrt{49},\ 0.\overline{3},\ -\dfrac{5}{8}$
4.
Which statement is true?
1. (A)
  $\sqrt{\dfrac{16}{25}}$ is irrational because it is a square root.
2. (B)
  $\pi = \dfrac{22}{7}$ , so $\pi$ is rational.
3. (C)
  $\sqrt{\dfrac{16}{25}} = \dfrac{4}{5}$ , so it is rational.
4. (D)
  $0.\overline{3}$ is irrational because its decimal never ends.
5.
How many integers $n$ satisfy $\sqrt{10} < n < \sqrt{90}$ ?
1. (A)
  $5$
2. (B)
  $6$
3. (C)
  $7$
4. (D)
  $8$
5. (E)
  $80$

Answer key — Real numbers

1.
7reals-radicals-sci-notation-01
$49 < 50 < 64$ gives $7 < \sqrt{50} < 8$ . The smaller whole number is $\mathbf{7}$ (and $\sqrt{50} \approx 7.07$ , just above it).
2.
4.56reals-radicals-sci-notation-02
$4{,}560{,}000 = 4.56 \times 10^6$ , so $a = \mathbf{4.56}$ . ( $45.6 \times 10^5$ is equal but is not scientific notation, because $45.6 \ge 10$ .)
3.
(D)
$\sqrt{49},\ 0.\overline{3},\ -\dfrac{5}{8}$
reals-radicals-sci-notation-03
$\sqrt{49} = 7$ , $0.\overline{3} = \frac{1}{3}$ , and $-\frac{5}{8}$ are all fractions of integers, so the last list is all rational: $\mathbf{\sqrt{49},\ 0.\overline{3},\ -\tfrac{5}{8}}$ . (Each other list contains an irrational: $\sqrt{50}$ ; $\pi$ — note $\frac{22}{7}$ and $3.14$ are rational approximations, not $\pi$ ; $\sqrt{2}$ and $\sqrt{8}$ .)
4.
(C)
$\sqrt{\dfrac{16}{25}} = \dfrac{4}{5}$ , so it is rational.
reals-radicals-sci-notation-04
$\sqrt{\frac{16}{25}} = \frac{4}{5}$ , a fraction of integers, so it is rational — the third statement is true. ( $\frac{22}{7}$ is only an approximation of $\pi$ ; $\pi$ is irrational. $0.\overline{3} = \frac{1}{3}$ is rational: every repeating decimal is a fraction.)
5.
(B)
$6$
reals-radicals-sci-notation-05
$3 < \sqrt{10} < 4$ and $9 < \sqrt{90} < 10$ , so $n$ can be $4, 5, 6, 7, 8, 9$ : $\mathbf{6}$ integers. ( $80 = 90 - 10$ counts the wrong thing; $7$ wrongly includes $3$ or $10$ , which lie outside the range.)