9. Maths Section - Exponents and Surds - ADF Aptitude Test Online Course - Exam Success

9. Maths Section - Exponents and Surds


This checkpoint has 2 parts (Part 1 - Exponents and Part 2 - Surds), each with their own video.

PART 1 - Exponents



An exponent is a power number.

For example, you might see 2³ (you may also see this written as 2^3). The top or last number is how many times you would multiply the first number by itself.

So, this is 2 x 2 x 2 = 8. So that's 2 multiplied by itself for 3 times. This is very different from multiplying 2 by the number three which gives you: 2 x 3 = 6.

“3” is an exponent because it is a little number at the top that tells us how many times to multiply the first number by itself.

So... they’re the basic rules you need to know for now.

You’ll learn more as we see it in action.

Let’s look at some questions now:

(x²)³(x⁶/x⁴) + 3x⁽¹²⁻⁴⁾ = ?
a) 4x⁸
b) x⁴
c) 3x²
d) x⁸

To answer this question, we need to apply some rules to make things easier. These rules are known as the index laws. The index laws you need to know are:

  • (xⁿ)(xⁱ) = x⁽ⁿ⁺ⁱ⁾
  • (xⁿ) / (xⁱ) = x⁽ⁿ⁻ⁱ⁾
  • (xⁿ)ⁱ = xⁿⁱ

To begin with, we need to simplify the term (x²)³. Since this is an index to the power of another index, we can multiply the two indices (2 x 3 = 6) to get a result of x⁶.

Next, we need to simplify the term (x⁶/x⁴). Since this is an index divided by another index, we need to subtract the second index (6 - 4 = 2) to get a result of x².

Our expression now reads (x⁶)(x²) + 3x⁽¹²⁻⁴⁾. The first two terms are multiplied together, meaning we can add their indices to get a product of x⁽⁶⁺²⁾ or x⁸. Then, we can simplify the index of the second term 12 - 4 to get 8, meaning that our expression becomes x⁸ + 3x⁸. Since both terms are to the same power of x, we can now add them as normal to get 1 + 3(x⁸) = 4x⁸ (a) - Remember if it doesn't have a number in front of the x or any other letter, it is 1, so x⁸ is that same as 1x⁸.

To recap:

  1. Use index laws to simplify equation
  2. If possible, get all values of x to the same power then simplify

Let's look at another question:

4 x 2 x 8² x 2 x 4² = 2^?

a) 6
b) 12
c) 14
d) 20

All of the terms in this equation (2, 4, and 8) are powers of base 2. In order to solve this question we need to get all terms to base 2.

First, we can consider that 4 = 2² and 8 = 2³. If we substitute this into the equation we then get:

2² x 2 x (2³)² x 2 x (2²)² = 2^?

Using the index law (xⁿ)ⁱ = xⁿⁱ so we know that this simplifies to:

2² x 2 x 2⁶ x 2 x 2⁴ = 2^?

Next, we use the index law (xⁿ)(xⁱ) = x⁽ⁿ⁺ⁱ⁾ to further simplify this equation.

Under this rule, 2⁽²⁺¹⁺⁶⁺¹⁺⁴⁾ = 2^?, meaning that ? = 2 + 1 + 6 + 1 + 4.

Adding up these numbers, we find that ? = 14. Therefore 4 x 2 x 8² x 2 x 4² = 2¹⁴.

One less effective way you might have tried to solve this question is by expanding the equation - that is to say, multiplying 4 by 2 by 8² and so on to get a result of 16384.

While this does eventually find you the answer, it means working with very large numbers, and it’s almost impossible to work out that 16384 = 2¹⁴ in your head.

It’s much easier to work with base 2 and indexes.

To recap:

  1. Put all values in the same base (e.g. 4 = 2²)
  2. Use index laws to simplify equation
  3. Solve for the required index value

PART 2 - Surds



What is a surd? A surd is a number (written in a special way) that cannot be simplified to remove a square root (or cube root etc).

What you normally see are typical numbers e.g. 4, 5, 4.5, -4, 5 - but beyond that there are special types of numbers that you have to express with certain symbols. A surd is one of them. A fun way to think of it is that there are a certain number of swimming strokes that you see - freestyle, butterfly etc... but then there are a whole range of odd-looking swimming stokes that people may invent and sometimes you can say that it's a certain stroke, but it is a stroke.

A square root is a number where its value, when multiplied by itself, gives the resulting number.

Example: 4 × 4 = 16, so a square root of 16 is 4.

Going back to surds, we know that 2 x 2 = 4 and that 2 is a square root of 4.

So… if you then write √4, this can be simplified to 2. So √4 is not a surd. But √2 is a surd because you cannot further simplify it.

So the number √2 is a surd (that's how you express the number) but if you do (√2)² you get...2

So... they’re the basic rules you need to know for now.

You’ll learn more as we see it in action.

Let’s look at some questions now:

√(36) / √(72)= ?
a) √(1/2)
b) √(3/2)
c) √(12)
d) 2

To answer this question, we need to simplify these two square roots. A square root means the number which when multiplied by itself gives the required value - for instance, if 2 x 2 = 4, then √4 =2

By the same principle, since 6 x 6 = 36, √(36) = 6. Our equation now reads 6 ÷ √(72) - Remember, the division sign can be shown as / or ÷.

Next, we need to simplify √(72).

Since 72 = 2 x 36, we can rewrite this as √(36) x √2, or 6√2.

Our equation now reads 6 ÷ 6√2, and by cancelling the 6 above and below the line we get a result of √(1/2) (a).

Another way we can solve this question is by simplifying √(36) / √(72) to √(36/72). Since 36/72 = ½, this also simplifies to √(1/2) or 1/√2 (since √1= 1).

To recap:

  1. Simplify all surds to their simplest form
  2. Cancel common factors

Here's another example:

Simplify √(72) - √(50)
a) √(22)
b) √(72/50)
c) 3√2
d) √2

This question similarly requires us to simplify these surds to their simplest form and then cancel any common factors.

We can start with √(72). Since this is equal to √(36)√(2), we know that it is then also equivalent to 6√2.

Then, we can simplify √(50).

In this case we know that √(25) = 5, so we can simplify √(50) to 5√2. Our equation then becomes 6√2-5√2.

Now that all our surds are the same base (√2), we can subtract them. Our expression reads (6-5)√2, which is the same as 1√2 or √2 (d). To recap:

  1. Make sure all surds have the same base
  2. Subtract 5√2 from 6√2

Here's another practice question.

If a box is full of 125 smaller cubes with side length 2cm, what is the volume of the box?
a) 250 cm³
b) 1000 cm³
c) 125 cm³
d) 62.25cm³

This question involves a mixture of geometry and indices, and can be solved in two different ways - using volume, or using side length.

To solve this question using side length we need to consider the number of small boxes which fit along each side of the larger box. Since the volume of the large box is x³ where x = side length, the number of smaller boxes along each side is ∛(125)= 5 boxes.

Since each smaller box has a side length of 2cm, we therefore know that the side lengths of the larger cube are 5 x 2 = 10cm.

We can then cube this side length to get V=10³cm³ =1000cm³.

Another way to solve this question is to use volume. We know that the side length of each smaller cube is 2cm, meaning that its volume is 2³cm³ or 8cm³.

Since there are 125 smaller boxes in the larger box, we can then multiply this volume by the number of boxes to find the volume of the larger box:

125 x 8 = 1000

Both of these methods give us a volume of 1000 cm³ (b). To recap:

  1. Find either the volume or side length of the smaller cube
  2. Apply to the number of smaller cubes in the larger cube
  3. Use this to find the volume of the larger cube

Now it's time to do your assignment - there are two assignments in this checkpoint:

  1. Download the assignment questions for exponents here and for surds here.
  2. Print it out or if you want to do it electronically, save it.
  3. Complete the questions to it.
  4. Then check the solutions on the video below. The answer key is also on the final page of your downloaded assignment questions.

Solutions Video for Exponents



Solutions Video for Surds




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