Exponents

High School Math 1 · Unit 1 · laws-of-exponents · Teacher edition

laws-of-exponents

Name ______________________________Date ______________

1.
Evaluate $8^{2/3}$ .
Answer: ______________
2.
Simplify $\dfrac{12x^{-3}y^{4}}{4x^{2}y^{-1}}$ . Write the result with positive exponents only.
Answer: ______________
3.
Light travels at $3 \times 10^{8}$ metres per second. The Sun is $1.5 \times 10^{11}$ metres from Earth. How many seconds does sunlight take to reach Earth?
Answer: ______________
4.
Which is the value of $\left(2^{3} \cdot 2^{-5}\right)^{2}$ ?
1. (A)
  $-16$
2. (B)
  $1$
3. (C)
  $\dfrac{1}{16}$
4. (D)
  $2^{-30}$
5.
If $\dfrac{4^{10} \cdot 8^{5}}{16^{6}} = 2^{n}$ , what is $n$ ?
1. (A)
  $5$
2. (B)
  $9$
3. (C)
  $10$
4. (D)
  $11$
5. (E)
  $35$

Answer key — Exponents

1.
4laws-of-exponents-01
$8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = \mathbf{4}$ . (Root first, then power: $\sqrt[3]{64} = 4$ gives the same thing, just with bigger numbers.)
2.
$\frac{3y^5}{x^5}$ laws-of-exponents-02
$\frac{12}{4} = 3$ , $\frac{x^{-3}}{x^{2}} = x^{-5}$ , $\frac{y^{4}}{y^{-1}} = y^{5}$ . So the quotient is $3x^{-5}y^{5}$ , and moving $x^{-5}$ downstairs gives $\mathbf{\dfrac{3y^{5}}{x^{5}}}$ .
3.
500laws-of-exponents-03
$\dfrac{1.5 \times 10^{11}}{3 \times 10^{8}} = 0.5 \times 10^{3} = \mathbf{500}$ seconds — a little over eight minutes.
4.
(C)
$\dfrac{1}{16}$
laws-of-exponents-04
$2^{3} \cdot 2^{-5} = 2^{-2}$ , and $(2^{-2})^2 = 2^{-4} = \dfrac{1}{2^4} = \mathbf{\dfrac{1}{16}}$ . ( $-16$ treats the negative exponent as a minus sign; $2^{-30}$ multiplies $3 \cdot (-5)$ instead of adding; $1$ comes from adding $-2 + 2$ instead of multiplying when squaring.)
5.
(D)
$11$
laws-of-exponents-05
$4^{10} = (2^2)^{10} = 2^{20}$ , $8^{5} = (2^3)^{5} = 2^{15}$ , $16^{6} = (2^4)^{6} = 2^{24}$ . Then $\dfrac{2^{20} \cdot 2^{15}}{2^{24}} = 2^{20 + 15 - 24} = 2^{11}$ , so $n = \mathbf{11}$ . ( $35$ forgets the denominator; $9$ uses the exponents $10 + 5 - 6$ without converting the bases.)