8. A banana skin makes up about $$ \frac{1}{8} $$ of the mas Robyn buys a bunch of bananas costing R22,55. The bananas when peeled have a mass of $$ 1,4 \mathrm{~kg} $$. a) Calculate the price per kilogram of the bananas when they are peeled. Give your answer to the nearest cent. 9. Determine the value of the following without using a calculator: c) $$ \left(2^{2} ight)^{3} $$ a) $$ 2^{2}+2^{3} $$ b) $$ 2^{2} imes 2^{3} $$ d) $$ 2^{3} imes 2^{-1} imes 2^{-4} $$ e) $$ 3^{4} imes 3^{\circ} $$ g) $$ 2^{-5} imes 2^{3} $$ h) $$ \left(2^{2} imes 3 ight)^{2} $$ f) $$ (5)^{-2} $$ 10. Simplify: a) $$ a^{-n} $$ b) $$ 2^{-1} $$ i) $$ 5^{0} imes 3^{-2} $$ d) $$ \left(4^{0} ight)^{-5} $$ e) $$ \left(\frac{1}{8} ight)^{2} $$ c) $$ 5^{2} imes 5^{-4} $$ 11. Simplify: a) $$ 10 x^{10} \div 5 x^{5} $$ b) $$ \left(2 x^{2} y^{3} z ight)^{3} $$ f) $$ \left(3^{2} ight)^{-4} $$ d) $$ \sqrt{\left(x^{4} ight)} imes x^{6} $$ e) $$ \frac{\sqrt{(64 b)^{4}}}{4^{2} b^{2}} $$ c) $$ (\sqrt{x})^{2} $$ 12. Write these numbers in their ordinary forms. a) $$ 7 imes 10^{5} $$ b) $$ 1,3 imes 10^{-2} $$ f) $$ \left(x^{3} ight)^{5} \div x^{6} $$ 13. Write in scientific notation: a) 45800000 b) 0,000052 c) 1,29 $$ imes 1 $$ 14. A chocolate company decides to increase the size of its chocolate bar in the ratio 7:6. If the old chocolate bar has a mass of 60 g , what will the mass of new bar be? 15. Lerato achieved 70 out of 200 in a test. What percentage is that? 16. You need vitamin $$ C $$ for the growth and repair of all the tissues in your body. Vitamin C deficiency causes scurvy, a disease that was common among sailors in the 16th and 17th centuries. The amount of ascorbic acid (vitamin C) found in a multi-vitamin tablet is $$ 5,0 imes 10^{-2} $$ grams. What is the total mass of ascorbic acid needed to make 60 multi-vitamin tablets?

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8. A banana skin makes up about $$ \frac{1}{8} $$ of the mas Robyn buys a bunch of bananas costing R22,55. The bananas when peeled have a mass of $$ 1,4 \mathrm{~kg} $$. a) Calculate the price per kilogram of the bananas when they are peeled. Give your answer to the nearest cent. 9. Determine the value of the following without using a calculator: c) $$ \left(2^{2}ight)^{3} $$ a) $$ 2^{2}+2^{3} $$ b) $$ 2^{2} 	imes 2^{3} $$ d) $$ 2^{3} 	imes 2^{-1} 	imes 2^{-4} $$ e) $$ 3^{4} 	imes 3^{\circ} $$ g) $$ 2^{-5} 	imes 2^{3} $$ h) $$ \left(2^{2} 	imes 3ight)^{2} $$ f) $$ (5)^{-2} $$ 10. Simplify: a) $$ a^{-n} $$ b) $$ 2^{-1} $$ i) $$ 5^{0} 	imes 3^{-2} $$ d) $$ \left(4^{0}ight)^{-5} $$ e) $$ \left(\frac{1}{8}ight)^{2} $$ c) $$ 5^{2} 	imes 5^{-4} $$ 11. Simplify: a) $$ 10 x^{10} \div 5 x^{5} $$ b) $$ \left(2 x^{2} y^{3} zight)^{3} $$ f) $$ \left(3^{2}ight)^{-4} $$ d) $$ \sqrt{\left(x^{4}ight)} 	imes x^{6} $$ e) $$ \frac{\sqrt{(64 b)^{4}}}{4^{2} b^{2}} $$ c) $$ (\sqrt{x})^{2} $$ 12. Write these numbers in their ordinary forms. a) $$ 7 	imes 10^{5} $$ b) $$ 1,3 	imes 10^{-2} $$ f) $$ \left(x^{3}ight)^{5} \div x^{6} $$ 13. Write in scientific notation: a) 45800000 b) 0,000052 c) 1,29 $$ 	imes 1 $$ 14. A chocolate company decides to increase the size of its chocolate bar in the ratio 7:6. If the old chocolate bar has a mass of 60 g , what will the mass of new bar be? 15. Lerato achieved 70 out of 200 in a test. What percentage is that? 16. You need vitamin $$ C $$ for the growth and repair of all the tissues in your body. Vitamin C deficiency causes scurvy, a disease that was common among sailors in the 16th and 17th centuries. The amount of ascorbic acid (vitamin C) found in a multi-vitamin tablet is $$ 5,0 	imes 10^{-2} $$ grams. What is the total mass of ascorbic acid needed to make 60 multi-vitamin tablets?

UpStudy AI · Accepted Answer

Let's solve the problems step by step.

### Problem 8: Price per kilogram of peeled bananas

**Known conditions:**
- Cost of bananas: R22.55
- Mass of peeled bananas: 1.4 kg

**Step 1: Calculate the price per kilogram.**
To find the price per kilogram, we divide the total cost by the mass of the bananas.

$$
\text{Price per kg} = \frac{\text{Total Cost}}{\text{Mass}} = \frac{22.55}{1.4}
$$

Now, let's calculate this value.

### Problem 9: Evaluate the expressions

**c)** $$ (2^{2})^{3} $$

Using the power of a power property $$ (a^{m})^{n} = a^{m \cdot n} $$:

$$
(2^{2})^{3} = 2^{2 \cdot 3} = 2^{6} = 64
$$

**a)** $$ 2^{2} + 2^{3} $$

Calculating each term:

$$
2^{2} = 4, \quad 2^{3} = 8
$$
$$
2^{2} + 2^{3} = 4 + 8 = 12
$$

**b)** $$ 2^{2} \times 2^{3} $$

Using the property $$ a^{m} \times a^{n} = a^{m+n} $$:

$$
2^{2} \times 2^{3} = 2^{2+3} = 2^{5} = 32
$$

**d)** $$ 2^{3} \times 2^{-1} \times 2^{-4} $$

Using the same property:

$$
2^{3} \times 2^{-1} \times 2^{-4} = 2^{3 - 1 - 4} = 2^{-2} = \frac{1}{4}
$$

**e)** $$ 3^{4} \times 3^{0} $$

Since $$ 3^{0} = 1 $$:

$$
3^{4} \times 3^{0} = 3^{4} = 81
$$

**g)** $$ 2^{-5} \times 2^{3} $$

Using the property:

$$
2^{-5} \times 2^{3} = 2^{-5 + 3} = 2^{-2} = \frac{1}{4}
$$

**h)** $$ (2^{2} \times 3)^{2} $$

Using the property $$ (ab)^{n} = a^{n}b^{n} $$:

$$
(2^{2} \times 3)^{2} = (2^{2})^{2} \times 3^{2} = 2^{4} \times 9 = 16 \times 9 = 144
$$

**f)** $$ (5)^{-2} $$

This is:

$$
(5)^{-2} = \frac{1}{5^{2}} = \frac{1}{25}
$$

### Problem 10: Simplify the expressions

**a)** $$ a^{-n} = \frac{1}{a^{n}} $$

**b)** $$ 2^{-1} = \frac{1}{2} $$

**i)** $$ 5^{0} \times 3^{-2} $$

Since $$ 5^{0} = 1 $$:

$$
5^{0} \times 3^{-2} = 1 \times \frac{1}{3^{2}} = \frac{1}{9}
$$

**d)** $$ (4^{0})^{-5} $$

Since $$ 4^{0} = 1 $$:

$$
(4^{0})^{-5} = 1^{-5} = 1
$$

**e)** $$ \left(\frac{1}{8}\right)^{2} = \frac{1^{2}}{8^{2}} = \frac{1}{64} $$

**c)** $$ 5^{2} \times 5^{-4} = 5^{2 - 4} = 5^{-2} = \frac{1}{25} $$

### Problem 11: Simplify the expressions

**a)** $$ 10 x^{10} \div 5 x^{5} = \frac{10}{5} \cdot x^{10 - 5} = 2 x^{5} $$

**b)** $$ (2 x^{2} y^{3} z)^{3} = 2^{3} x^{6} y^{9} z^{3} = 8 x^{6} y^{9} z^{3} $$

**f)** $$ (3^{2})^{-4} = 3^{-8} = \frac{1}{3^{8}} $$

**d)** $$ \sqrt{(x^{4})} \times x^{6} = x^{2} \times x^{6} = x^{8} $$

**e)** $$ \frac{\sqrt{(64 b)^{4}}}{4^{2} b^{2}} = \frac{(64 b)^{2}}{16 b^{2}} = \frac{64^{2} b^{2}}{16 b^{2}} = \frac{64^{2}}{16} = 256 $$

**c)** $$ (\sqrt{x})^{2} = x $$

### Problem 12: Write in ordinary forms

**a)** $$ 7 \times 10^{5} = 700000 $$

**b)** $$ 1.3 \times 10^{-2} = 0.013 $$

**f)** $$ \left(x^{3}\right)^{5} \div x^{6} = x^{15} \div x^{6} = x^{15 - 6} = x^{9} $$

### Problem 13: Write in scientific notation

**a)** $$ 45800000 = 4.58 \times 10^{7} $$

**b)** $$ 0.000052 = 5.2 \times 10^{-5} $$

**c)** $$ 1.29 \times 1 = 1.29 $$

### Problem 14: Mass of the new chocolate bar

**Known conditions:**
- Old mass: 60 g
- Ratio: 7:6

**Step 1: Calculate the new mass.**
Let the new mass be $$ x $$.

$$
\frac{x}{60} = \frac{7}{6}
$$

Cross-multiplying gives:

$$
6x = 420 \implies x = 70 \text{ g}
$$

### Problem 15: Percentage achieved by Lerato

**Known conditions:**
- Marks obtained: 70
- Total marks: 200

**Step 1: Calculate the percentage.
**Problem 8:** - **Price per kilogram of peeled bananas:** R16.10 **Problem 9:** - **c)** 64 - **a)** 12 - **b)** 32 - **d)** 0.25 - **e)** 81 - **g)** 0.25 - **h)** 144 - **f)** 0.04 **Problem 10:** - **a)** $$ \frac{1}{a^{n}} $$ - **b)** 0.5 - **i)** $$ \frac{1}{9} $$ - **d)** 1 - **e)** $$ \frac{1}{64} $$ - **c)** $$ \frac{1}{25} $$ **Problem 11:** - **a)** $$ 2x^{5} $$ - **b)** $$ 8x^{6}y^{9}z^{3} $$ - **f)** $$ \frac{1}{3^{8}} $$ - **d)** $$ x^{8} $$ - **e)** 256 - **c)** $$ x $$ **Problem 12:** - **a)** 700,000 - **b)** 0.013 - **f)** $$ x^{9} $$ **Problem 13:** - **a)** $$ 4.58 \times 10^{7} $$ - **b)** $$ 5.2 \times 10^{-5} $$ - **c)** 1.29 **Problem 14:** - **Mass of new chocolate bar:** 70 g **Problem 15:** - **Percentage achieved by Lerato:** 35% **Problem 16:** - **Total mass of ascorbic acid needed:** 0.3 grams

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