Year 9 Indices and surds: free practice test with answers

Sample indices and surds questions with answers

Try each one before you open the answer.

1. Index notation

   Evaluate $2^{3} \times 3^{2}$.

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   Key idea: Work out each power first, then multiply.

   1. $2^{3} = 2 \times 2 \times 2 = 8$
   2. $3^{2} = 3 \times 3 = 9$
   3. $8 \times 9 = 72$

   Answer: $72$

2. Index laws for multiplication and division

   Simplify each expression.

   (a) $x^3 \times x^5$

   (b) $\dfrac{y^8}{y^3}$

   (c) $3a^4 \times 2a^2$

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   Key idea: When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.

   1. (a) $x^3 \times x^5 = x^{3+5} = x^8$
   2. (b) $\dfrac{y^8}{y^3} = y^{8-3} = y^5$
   3. (c) Multiply the coefficients and apply the product law to the variable: $3a^4 \times 2a^2 = (3 \times 2)a^{4+2} = 6a^6$

   Answer: (a) $x^8$ (b) $y^5$ (c) $6a^6$

3. Index laws for power of a power and the zero index

   Simplify each expression.

   (a) $(m^3)^4$

   (b) $(2p^2)^3$

   (c) $5x^0 + (3y)^0$

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   Key idea: For a power of a power, multiply the exponents. Any non-zero base raised to the power of zero equals 1.

   1. (a) $(m^3)^4 = m^{3 \times 4} = m^{12}$
   2. (b) Apply the power to both the coefficient and the variable: $(2p^2)^3 = 2^3 \times p^{2 \times 3} = 8p^6$
   3. (c) $x^0 = 1$ and $(3y)^0 = 1$, so $5x^0 + (3y)^0 = 5(1) + 1 = 6$

   Answer: (a) $m^{12}$ (b) $8p^6$ (c) $6$

4. Index laws for brackets and fractions

   A rectangular garden bed has a length of $4b^4$ metres and a width of $3b$ metres.

   (a) Write a simplified expression for the area of the garden bed in square metres.

   (b) A second garden bed has area $(2b^2)^3$ square metres. Write a simplified expression for how many times larger the second garden bed's area is than the first garden bed's area.

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   Key idea: Find each area using the exponent laws, then divide to find the ratio.

   1. (a) Area of first garden bed: $4b^4 \times 3b = 12b^5$ m²
   2. (b) Area of second garden bed: $(2b^2)^3 = 2^3 \times b^{2 \times 3} = 8b^6$ m²
   3. Divide the second area by the first:\

   $$\frac{8b^6}{12b^5} = \frac{8}{12} \times b^{6-5} = \frac{2}{3}b$$

   4. The second garden bed's area is $\dfrac{2b}{3}$ times that of the first.

   Answer: (a) $12b^5$ m² (b) $\dfrac{2b}{3}$ times larger

5. Negative indices

   Write each expression using positive indices only.

   (a) $5^{-2}$ (b) $x^{-3}$ (c) $3a^{-4}$

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   Key idea: A negative index means a reciprocal: $a^{-n} = \frac{1}{a^n}$. The index applies only to the base it is attached to.

   1. (a) $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
   2. (b) $x^{-3} = \frac{1}{x^3}$
   3. (c) Only $a$ has the negative index, so $3a^{-4} = 3 \times \frac{1}{a^4} = \frac{3}{a^4}$

   Answer: (a) $\frac{1}{5^2} = \frac{1}{25}$ (b) $\frac{1}{x^3}$ (c) $\frac{3}{a^4}$

6. Scientific notation

   Write each number in the form requested.

   (a) Write $3.07 \times 10^5$ as an ordinary number.

   (b) Write $0.000 \, 048$ in scientific notation.

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   Key idea: In scientific notation a number is written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer. A positive power of 10 makes the number larger; a negative power makes it smaller.

   1. (a) $3.07 \times 10^5$ means move the decimal point 5 places to the right: $3.07 \times 10^5 = 307\,000$
   2. (b) The first significant digit in $0.000\,048$ is 4. Count how many places the decimal point must move right to give a number between 1 and 10: it moves 5 places, giving $4.8$. Because the original number is less than 1, the power is negative: $0.000\,048 = 4.8 \times 10^{-5}$

   Answer: (a) $307\,000$ and (b) $4.8 \times 10^{-5}$

7. Scientific notation using significant figures

   The distance from Earth to the Sun is approximately $1.496 \times 10^{11}$ metres. The distance from Earth to the Moon is approximately $3.84 \times 10^{8}$ metres. How many times farther from Earth is the Sun than the Moon? Give your answer correct to the nearest whole number.

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   Key idea: "How many times farther" means dividing the larger distance by the smaller distance. Divide numbers in scientific notation by handling the coefficients and powers of 10 separately.

   1. Set up the division:\

   $$\frac{1.496 \times 10^{11}}{3.84 \times 10^{8}}$$

   2. Divide the coefficients:\

   $$\frac{1.496}{3.84} \approx 0.38958\ldots$$

   3. Divide the powers of 10:\

   $$10^{11} \div 10^{8} = 10^{3}$$

   4. Combine and simplify:\

   $$0.38958\ldots \times 10^{3} = 389.58\ldots$$

   5. Round to the nearest whole number: $\approx 390$.

   Answer: The Sun is approximately $390$ times farther from Earth than the Moon.

8. Fractional indices and surds

   Evaluate without a calculator.

   (a) $49^{\frac{1}{2}}$

   (b) $27^{\frac{1}{3}}$

   (c) $16^{\frac{3}{4}}$

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   Key idea: $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = (\sqrt[n]{a})^{m}$.

   1. $49^{\frac{1}{2}} = \sqrt{49} = 7$
   2. $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$
   3. $16^{\frac{3}{4}} = (\sqrt[4]{16})^{3} = 2^{3} = 8$

   Answer: (a) $7$ (b) $3$ (c) $8$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 9 exponent laws worksheet
• Year 9 scientific notation worksheet