Exponents and Scientific Notation: practice solutions, Grade 8 – download the PDF

1 Parts of a power ★★★

1. The base is \(6\) and the exponent is \(4\).
2. \(6^4 = 6 \times 6 \times 6 \times 6\).
3. \(6 \times 6 = 36\), \(36 \times 6 = 216\), \(216 \times 6 = 1{,}296\). So \(6^4 = 1{,}296\).

2 Evaluating powers ★★★

\(2^6 = 64\) ; \(10^3 = 1{,}000\) ; \((-2)^3 = -8\) (odd exponent, negative result) ; \((-4)^2 = 16\) (the base \(-4\) is squared) ; \(-4^2 = -(4 \times 4) = -16\) (only the 4 is squared) ; \(3^4 = 81\).

3 Using the product rule ★★★

Same base, so add the exponents. \(4^3 \times 4^5 = 4^8\) ; \(9 \times 9^6 = 9^1 \times 9^6 = 9^7\) ; \(x^2 \times x^7 = x^9\) ; \(y^5 \times y = y^{5+1} = y^6\).

4 Using the quotient rule ★★★

Same base, so subtract the exponents. \(\dfrac{6^9}{6^4} = 6^5\) ; \(\dfrac{10^8}{10^3} = 10^5\) ; \(\dfrac{z^{12}}{z^5} = z^7\) ; \(\dfrac{2^7}{2^7} = 2^0 = 1\), which matches the fact that any nonzero number divided by itself is 1.

5 Powers of powers ★★★

Multiply the exponents. \((3^2)^4 = 3^8\) ; \((5^3)^3 = 5^9\) ; \((y^6)^2 = y^{12}\) ; \((10^2)^5 = 10^{10}\).

6 Zero and negative exponents ★★★

\(7^0 = 1\) ; \(3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}\) ; \(10^{-3} = \dfrac{1}{1{,}000} = 0.001\) ; \(2^{-4} = \dfrac{1}{16}\) ; \((-5)^0 = 1\) (any nonzero base to the power 0).

7 Standard form ★★★

\(3.2 \times 10^4 = 32{,}000\) (4 places right) ; \(7 \times 10^6 = 7{,}000{,}000\) ; \(5.1 \times 10^{-3} = 0.0051\) (3 places left) ; \(9.04 \times 10^2 = 904\).

8 True or false? ★★★

1. False. Add the exponents: \(2^3 \times 2^4 = 2^7 = 128\), not \(2^{12}\).
2. True. Multiply the exponents: \((3^2)^3 = 3^{2 \times 3} = 3^6 = 729\).
3. False. A negative exponent gives a reciprocal: \(5^{-1} = \dfrac{1}{5}\).
4. False. Any nonzero base to the power 0 equals 1, so \(4^0 = 1\).

9 Which is bigger? ★★★

(a) \(2^5 = 32\) and \(5^2 = 25\), so \(2^5\) is greater.

(b) \(3^4 = 81\) and \(4^3 = 64\), so \(3^4\) is greater.

(c) \(2^{10} = 1{,}024\) and \(10^2 = 100\), so \(2^{10}\) is greater. Swapping base and exponent does not give the same value!

10 Simplifying with variables ★★★

(a) Multiply the numbers and add the exponents: \(4 \times 5 = 20\), so \(20x^{3+4} = 20x^7\).

(b) \(\dfrac{18}{6} = 3\) and \(m^{9-4} = m^5\), so the result is \(3m^5\).

(c) \((2a^2)^3 = 2^3 \times (a^2)^3 = 8a^6\).

11 To scientific notation ★★★

\(58{,}000{,}000 = 5.8 \times 10^7\) (7 places left) ; \(0.00071 = 7.1 \times 10^{-4}\) (4 places right) ; \(640{,}000{,}000{,}000 = 6.4 \times 10^{11}\) (11 places left) ; \(0.0000093 = 9.3 \times 10^{-6}\) (6 places right).

12 Ordering numbers ★★★

1. Compare exponents first, then decimal parts: \(8.1 \times 10^4\) (exponent 4) is least. Next, with exponent 5: \(4.2 \times 10^5 < 6 \times 10^5\). Greatest is \(3.9 \times 10^6\). Order: \(8.1 \times 10^4 < 4.2 \times 10^5 < 6 \times 10^5 < 3.9 \times 10^6\).
2. The values are \(0.0072\), \(0.0009\) and \(0.0011\). Order: \(9 \times 10^{-4} < 1.1 \times 10^{-3} < 7.2 \times 10^{-3}\).

13 Multiplying in scientific notation ★★★

(a) \(2.5 \times 4 = 10\) and \(10^3 \times 10^5 = 10^8\), so the product is \(10 \times 10^8 = 1 \times 10^9\).

(b) \(6 \times 3 = 18\) and \(10^{-2} \times 10^7 = 10^5\), so the product is \(18 \times 10^5 = 1.8 \times 10^6\).

14 Phone storage ★★★

1. \(256 \times 10^9 = 2.56 \times 10^2 \times 10^9 = 2.56 \times 10^{11}\) bytes.
2. \(\dfrac{2.56 \times 10^{11}}{2 \times 10^3} = 1.28 \times 10^{8}\). About \(128{,}000{,}000\) text files would fit.

15 Simplifying with negative exponents ★★★

1. Add the exponents on top: \(x^{3-7} = x^{-4}\). Then subtract: \(x^{-4-(-2)} = x^{-2} = \dfrac{1}{x^2}\).
2. Top: \(2^{-3+5} = 2^2\). Then \(2^{2-(-1)} = 2^3 = 8\).
3. \((5^2)^{-2} = 5^{-4}\), then \(5^{-4} \times 5^3 = 5^{-1} = \dfrac{1}{5}\).

16 Reading an exponential curve ★★★

1. \(2^2 = 4\), so \(y = 4\). And \(2^{-1} = \dfrac{1}{2}\), so \(y = 0.5\).
2. \(2^3 = 8\), so \(x = 3\). \(2^{-1} = \dfrac{1}{2}\), so \(x = -1\).
3. A power of 2 is never zero and never negative: \(2^{-n} = \dfrac{1}{2^n}\) gets smaller and smaller but stays positive. The curve gets closer to the axis without reaching it.

17 Doubling bacteria ★★★

1. \(500 \times 2^6 = 500 \times 64 = 32{,}000\) bacteria.
2. We need \(500 \times 2^n > 10^6\), that is \(2^n > 2{,}000\). Since \(2^{10} = 1{,}024\) is too small and \(2^{11} = 2{,}048\) works, the answer is \(n = 11\). Check: after 10 hours there are \(512{,}000\); after 11 hours there are \(1{,}024{,}000\). The population first exceeds one million after 11 hours.

18 Adding and subtracting ★★★

(a) Rewrite \(8.5 \times 10^4 = 0.85 \times 10^5\). Then \(6.2 \times 10^5 + 0.85 \times 10^5 = 7.05 \times 10^5\).

(b) Rewrite \(2.6 \times 10^{-4} = 0.26 \times 10^{-3}\). Then \(4.1 \times 10^{-3} - 0.26 \times 10^{-3} = 3.84 \times 10^{-3}\).

19 How long does light take? ★★★

Time = distance \(\div\) speed.

Sun: \(\dfrac{1.5 \times 10^{11}}{3 \times 10^8} = 0.5 \times 10^3 = 5 \times 10^2 = 500\) s, which is \(8\) min \(20\) s.

Moon: \(\dfrac{3.84 \times 10^8}{3 \times 10^8} = 1.28\) s.

Sunlight takes about 8 minutes 20 seconds to reach Earth; moonlight takes about 1.28 seconds.

20 How many times larger? ★★★

Since \(8 > 6\), the country is larger. Divide: \(\dfrac{3.3 \times 10^8}{2.4 \times 10^6} = \dfrac{3.3}{2.4} \times 10^2 = 1.375 \times 10^2\).

Rounded: \(1.4 \times 10^2\). The country has about 140 times as many residents as the city.