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Solutions
29 Videos

A bacterial colony with an initial population of 300 doubles every day. Write the equation models for this exponential growth.

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Q1

What is the value of a non-zero number raised to the exponent zero?

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Q3

A radioactive substance with an initial mass of 250 mg has a half-life of 1 year.

a) Write an equation to relate the mass of radioactive material remaining to time.

b) What mass will remain after 10 years?

c) How long will it take for the sample to decay to 20% of its initial mass? Explain how you arrived at your answer.

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Q4

Evaluate. Express as a fraction in lowest terms.

a) ```
\displaystyle
10^{-1}
```

b) ```
\displaystyle
4^{-2}
```

c) ```
\displaystyle
3^{-2} +9^{-1}
```

d) ```
\displaystyle
5^{-3} +5^0
```

e) ```
\displaystyle
(\frac{1}{5})^{-1}
```

f) ```
\displaystyle
(\frac{3}{4})^{-3}
```

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Q6

Simplify.

```
\displaystyle
(x^{-2})(x^{-1})(x^6)
```

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Q7a

Simplify.

```
\displaystyle
(3km^2)(2k^{-2}m^{-2})
```

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Q7b

Simplify.

```
\displaystyle
w^{-3} \div w^{-2}
```

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Q7c

Simplify.

```
\displaystyle
\frac{u^{-2}v^3}{u^{-3}v^{-2}}
```

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Q7d

Simplify.

```
\displaystyle
(z^{-3})^{-2}
```

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Q7e

Simplify.

```
\displaystyle
(2ab^{-1})^{-2}
```

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Q7f

Evaluate ```
\displaystyle
\sqrt[3]{64}
```

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Q8a

Evaluate ```
\displaystyle
\sqrt[4]{625}
```

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Q8b

Evaluate ```
\displaystyle
\sqrt[5]{-3125}
```

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Q8c

Evaluate ```
\displaystyle
(\frac{1}{64})^{\frac{1}{6}}
```

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Q8d

Evaluate ```
\displaystyle
27^{\frac{2}{3}}
```

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Q8e

Evaluate ```
\displaystyle
(-1000)^{\frac{4}{3}}
```

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Q8f

Evaluate ```
\displaystyle
-4^{-3}
```

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Q8g

Evaluate ```
\displaystyle
(\frac{3}{4})^{-2}
```

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Q8h

Evaluate ```
\displaystyle
(- \frac{27}{125})^{- \frac{2}{3}}
```

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Q8i

The length, `x`

, in centimetres, by which a spring with spring constant `k`

is stretched
or compressed from its test position is related to its stored potential energy, `U`

, in
joules (`J`

), according to the equation

```
\displaystyle
x = (2Uk^{-1})^{\frac{1}{2}}
```

a) Use the power of a power rule to write this equation in a different form.

b) Write the equation in radical form, using a single radical.

c) A spring with spring constant 10 has 320 J of stored energy. By how much is this spring stretched?

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Q9

a) Graph the function ```
\displaystyle
y = 27(\frac{1}{3})^x
```

b) Identify the

- i) domain, ii) range
- iii) x and y intercepts, if they exist
- iv) intervals of increase and decrease
- v) equation of the asymptote

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Q10

Determine the equation for the exponential graph shown.

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0.46mins

Q11

a) Sketch the function `y = 2^{x-3}+ 4`

.

b) Identify the

- i) domain
- ii) range
- iii) equation of the asymptote

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Q12

Describe the transformation or transformations that map the base function `y = 5^x`

onto each given function.

```
\displaystyle
y = 2(5^x)
```

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Q13a

Describe the transformation or transformations that map the base function `y = 5^x`

onto each given function.

```
\displaystyle
y = 5^{2x}
```

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Q13b

Describe the transformation or transformations that map the base function `y = 5^x`

onto each given function.

```
\displaystyle
y = -5^{-x}
```

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Q13c

`y = 5^x`

onto each given function.

```
\displaystyle
y = -5^{-5x -10}
```

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Q13d

The height, `h`

, in centimetres, of a bouncing ball after `n`

bounces is given.

a) Calculate the first and second differences and describe the trend.

b) Make a scatter plot of height versus number of bounces. Describe the shape of the curve.

c) Perform an appropriate regression analysis on the data. Write the equation of the curve of best fit. Justify your choice of the type of regression curve.

d) Will the ball ever stop bouncing? Discuss this with respect to

- i) the mathematical model
- ii) the real situation

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0.39mins

Q14a