Solve Exponential Lesson

Presentation

•

Mathematics

•

7th - 10th Grade

•

Hard

Joseph Anderson

FREE Resource

12 Slides • 13 Questions

Exponential Functions and Logarithms

Year 11 Methods

Introduction

Exponential growth and decay are found in many practical situations, such as, finance, medicine, chemistry, earthquakes, computing, astronomy, sound, sports and marketing.
Mathematics defines exponential functions as $f\left(x\right)=ka^x$
where $k$ is a non-zero constant and $a$ is a positive real number other than 1

Exponents

Notice that index, power and exponent are interchangeable and that there is a powerful value difference between the operations:

2+4 = 6

2x4=8

2^4=16

Understanding Exponents

For more complex problems, you will need to expand the index form to find similar bases.

Multiple Choice

$12a^3b^2$ in expanded form is:

$12\times a\times a\times a\times b\times b$

$12\times3\times2\times a\times b$

Multiple Choice

What is the prime factorisation of 24?

$2\times12$

$2\times2\times3$

$2^3\times3$

Index Laws

The index laws help simplify expressions and equations.

When using the laws, it is important that you:

- only use them when the bases are the same and

- that you do not multiply/divide, etc the bases.

Multiple Choice

Simplify the following: a⁰ + a⁰

a⁰

2a⁰

Multiple Choice

a³ x a⁵

a¹⁵

a²

a⁸

a^-2

Multiple Choice

c⁵ ÷ c²

c⁷

c³

c¹⁰

c^-2

Multiple Choice

(y²)³

y⁵

y²³

y⁶

Multiple Choice

Simplify: $\frac{a^5b^{10}}{a^3b^3}$

$a^8b^{13}$

$a^2b^7$

$a^{15}b^{30}$

Multiple Choice

Simplify: (4pq³)² ÷ (2p²q)³

2p^-4q³

8p⁴q³

16p⁴q²

4p^-4q²

Multiple Choice

Simplify: $\frac{b^{5x}\times b^{2x+1}}{b^{3x}}$

$b^{4x+1}$

$b^{5x}$

Negative Indices

Using the second index law we can see that a negative indices are equal to the reciprocal with a positive index.
Always state the final answer as a positive index, unless otherwise stated.

Negative Index Law

Watch this video for conceptual understanding of the negative index law.

Multiple Choice

Simplify:
$\left(3ab^2\right)\left(-2b^{-3}\right)$

$-\frac{6a}{b}$

$-\frac{6a^2}{b}$

$ab$

$\frac{3a}{2b}$

Multiple Choice

Which of the following is equivalent to the expression $\frac{1}{2x^7}$ ?

$\left(2x\right)^{-7}$

$\frac{1}{2}x^7$

$\frac{1}{2}x^{-7}$

$2x^{-7}$

Multiple Choice

(y⁹)^-2

y¹⁸

y¹¹

1/y¹⁸

y⁹/2

Multiple Choice

How else could you write $4^{-2}$ ?

$16$

$-16$

$-\frac{1}{16}$

$\frac{1}{16}$

Composite Bases

Composite bases can be factorised into primes to simplify the calculations.

Check out the video on prime factorisation to recap!

Simplifying using prime factorisation

Consider the folllowing expressions:

\frac{12^3}{4^3}

(Using prime factorisation and the index laws we could simplify)

\frac{\left(3\times4\right)^3}{4^3}

\frac{3^3\times4^3}{4^3}

3^3=27

Exponential Functions and Logarithms

Year 11 Methods

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