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

1 Parts of a power ★★★

Consider \(6^4\).

1. Name the base and the exponent.
2. Write \(6^4\) as a repeated multiplication.
3. Calculate its value.

2 Evaluating powers ★★★

Evaluate each expression: \(2^6\) ; \(10^3\) ; \((-2)^3\) ; \((-4)^2\) ; \(-4^2\) ; \(3^4\).

3 Using the product rule ★★★

Write each product as a single power: \(4^3 \times 4^5\) ; \(9 \times 9^6\) ; \(x^2 \times x^7\) ; \(y^5 \times y\).

4 Using the quotient rule ★★★

Write each quotient as a single power: \(\dfrac{6^9}{6^4}\) ; \(\dfrac{10^8}{10^3}\) ; \(\dfrac{z^{12}}{z^5}\) ; \(\dfrac{2^7}{2^7}\).

5 Powers of powers ★★★

Simplify: \((3^2)^4\) ; \((5^3)^3\) ; \((y^6)^2\) ; \((10^2)^5\).

6 Zero and negative exponents ★★★

Evaluate: \(7^0\) ; \(3^{-2}\) ; \(10^{-3}\) ; \(2^{-4}\) ; \((-5)^0\). Give \(10^{-3}\) also as a decimal.

7 Standard form ★★★

Write each number in standard form (no power of 10): \(3.2 \times 10^4\) ; \(7 \times 10^6\) ; \(5.1 \times 10^{-3}\) ; \(9.04 \times 10^2\).

8 True or false? ★★★

Decide whether each statement is true or false, and justify.

1. \(2^3 \times 2^4 = 2^{12}\)
2. \((3^2)^3 = 3^6\)
3. \(5^{-1} = -5\)
4. \(4^0 = 0\)

9 Which is bigger? ★★★

Compute both sides and say which is greater: (a) \(2^5\) or \(5^2\) ; (b) \(3^4\) or \(4^3\) ; (c) \(2^{10}\) or \(10^2\).

10 Simplifying with variables ★★★

Simplify each expression: (a) \((4x^3)(5x^4)\) ; (b) \(\dfrac{18m^9}{6m^4}\) ; (c) \((2a^2)^3\).

11 To scientific notation ★★★

Write in scientific notation: \(58{,}000{,}000\) ; \(0.00071\) ; \(640{,}000{,}000{,}000\) ; \(0.0000093\).

12 Ordering numbers ★★★

Order each list from least to greatest.

1. \(4.2 \times 10^5\), \(3.9 \times 10^6\), \(8.1 \times 10^4\), \(6 \times 10^5\)
2. \(7.2 \times 10^{-3}\), \(9 \times 10^{-4}\), \(1.1 \times 10^{-3}\)

13 Multiplying in scientific notation ★★★

Compute and give each answer in scientific notation: (a) \((2.5 \times 10^3)(4 \times 10^5)\) ; (b) \((6 \times 10^{-2})(3 \times 10^7)\).

14 Phone storage ★★★

A phone has \(256\) GB of storage, where \(1\) GB \(= 10^9\) bytes. A text file takes about \(2 \times 10^3\) bytes.

1. Write the storage in bytes using scientific notation.
2. How many such text files could fit, assuming nothing else is stored?

15 Simplifying with negative exponents ★★★

Simplify, and write each answer with positive exponents only:

1. \(\dfrac{x^3 \times x^{-7}}{x^{-2}}\)
2. \(\dfrac{2^{-3} \times 2^5}{2^{-1}}\)
3. \((5^2)^{-2} \times 5^3\)

16 Reading an exponential curve ★★★

The graph shows \(y = 2^x\). Use it and the rules of exponents to answer.

1. What is the value of \(y\) when \(x = 2\)? when \(x = -1\)?
2. For which \(x\) does \(2^x = 8\)? \(2^x = \dfrac{1}{2}\)?
3. Why does the curve never touch the \(x\)-axis?

17 Doubling bacteria ★★★

A culture starts with \(500\) bacteria and the population doubles every hour, so after \(n\) hours there are \(500 \times 2^n\) bacteria.

1. How many bacteria are there after \(6\) hours?
2. After how many whole hours does the population first exceed one million?

18 Adding and subtracting ★★★

Compute and write the result in scientific notation: (a) \(6.2 \times 10^5 + 8.5 \times 10^4\) ; (b) \(4.1 \times 10^{-3} - 2.6 \times 10^{-4}\).

19 How long does light take? ★★★

Light travels at about \(3 \times 10^8\) meters per second. Find the time it needs to travel from the Sun to Earth (\(1.5 \times 10^{11}\) m) and from the Moon to Earth (\(3.84 \times 10^8\) m). Convert the first time into minutes and seconds.

20 How many times larger? ★★★

A city has about \(2.4 \times 10^6\) residents and a country has about \(3.3 \times 10^8\) residents. Which is larger, and about how many times larger? Give the ratio in scientific notation, rounded to two significant digits.