Indices

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ITEM	TYPE	NUMBER
Working with indices	Workout	45 slides
Indices	Library	26 questions

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SAMPLE FROM THE WORKOUT

Showing American English version

SLIDE 1 - QUESTION 1

SLIDE 2 - SOLUTION

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SAMPLE FROM THE LIBRARY

Showing American English version

QUESTION [difficulty 0.4]

SOLUTION

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DEPENDENCIES

		248: Working with space

		250: Indices

		254: Investments

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CONCEPTS

ITEM

LEV.

Negative index

556.3

Roots of powers and powers of roots

556.3

Calculator: exponent button

556.7

Base and index

556.9

Index rule: multiplying numbers to the same base

556.9

Negative index is reciprocal

557.1

Index rule: dividing numbers to the same base

557.3

Index rule: power of power

557.5

Simplifying prime powers

557.7

Simplifying algebraic indices

557.9

Fractional / rational index

558.3

Powers of rational indices

558.6

Algebraic rational indices

558.8

Volume to surface area ratio problem

559.0

Scaling of similar shapes problem

559.2

Valid / invalid

559.9

RAW CONTENT OF THE WORKOUT

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SLIDE 1 Revision Work out ? ? ? ? ? ? ? ? SLIDE 2 ? ? ? ? ? ? ? ? SLIDE 3 Revision Work out ? ? ? ? ? ? ? ? SLIDE 4 ? ? ? ? ? ? ? ? SLIDE 5 Revision Work out, giving your answer in standard form ? ? ? ? ? ? ? ? SLIDE 6 ? ? ? ? ? ? ? ? SLIDE 7 Big numbers Very big and very small numbers often appear in science Use the exponent button on your calculator to compute ? ? ? ? ? SLIDE 8 ? ? ? ? ? SLIDE 9 Example When multiplying indices to the same base, we add the indices Work out, giving your answer in index form ? ? ? ? SLIDE 10 ? ? ? ? SLIDE 11 Negative index corresponds to dividing Evaluate, expressing your answer as a fraction in its lowest form ? ? ? ? ? ? SLIDE 12 ? ? ? ? ? ? SLIDE 13 Dividing to the same base ? subtract indices Any index to the power 0 is equal to 1 Evaluate ? ? ? ? ? ? ? ? SLIDE 14 ? ? ? ? ? ? ? ? SLIDE 15 Powers of powers We often use a dot instead of a cross to indicate multiplication. If we place numbers together, that also means multiplication. Following the examples, write each of these in terms of their prime base ? ? ? ? ? ? SLIDE 16 ? ? ? ? ? ? SLIDE 17 Simplifying by prime factors Evaluate ? ? ? ? ? ? SLIDE 18 ? ? ? ? ? ? SLIDE 19 Simplify ? ? ? ? ? ? SLIDE20 ? ? ? ? ? ? SLIDE 21 Simplify ? ? ? ? ? ? SLIDE 22 ? ? ? ? ? ? SLIDE 23 The meaning of a fractional index A fraction is also called a rational number. A rational index means a root SLIDE 24 Evaluate ? ? ? ? ? ? SLIDE 25 ? ? ? ? ? ? SLIDE 26 Evaluate ? ? ? ? ? ? SLIDE 27 ? ? ? ? ? ? SLIDE 28 Simplify ? ? ? ? ? ? SLIDE 29 ? ? ? ? ? ? SLIDE 30 Two cubes, A and B, are such that the ratio of the surface area of A to the surface area of B is . If the volume of B is 10 cm3, find the volume of A. Give your answer to 3 significant figures. SLIDE 30B Two cubes, A and B, are such that the ratio of the surface area of A to the surface area of B is . If the volume of B is 10 in3, find the volume of A. Give your answer to 3 significant figures. SLIDE 31 Two cubes, A and B, are such that the ratio of the surface area of A to the surface area of B is . If the volume of B is 10 cm3, find the volume of A. Give your answer to 3 significant figures. Solution � numerical approach Alternative solution Observations SLIDE 31B Solution � numerical approach Alternative solution Observations SLIDE 32 Two shapes are similar if the one is a scaling of the other. All dimensions are multiplied by the same scale factor. Two shapes, A and B are similar such that the ratio of the base of A to the base of B is . ? What is the ratio of the surface area of A to the surface area of B? ? What is the ratio of the volume of A to the volume of B? SLIDE 33 Two shapes, A and B are similar such that the ratio of the base of A to the base of B is . ? What is the ratio of the surface area of A to the surface area of B? The actual shape does not matter. The surfaces of A and B may be thought of as a flat rectangle � or square � where the length and width are both in the ratio . So, the ratio of the surface areas is ? What is the ratio of the volume of A to the volume of B? The same argument applies where the volumes are treated as cubes. Thus, SLIDE 34 Two shapes, A and B are similar such that the ratio of the surface area of A to the surface area of B is . What, in terms of K, is the ratio of the volume of A to the volume of B? SLIDE 35 Two shapes, A and B are similar such that the ratio of the surface area of A to the surface area of B is . What, in terms of K, is the ratio of the volume of A to the volume of B? Solution Given the surface area of an object to find its volume, you take the square root and then cube that. Note also, SLIDE 36 The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of A is 8 cm2, what is the volume of B? Give your answer to 3 significant figures. SLIDE 37 The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of A is 8 cm2, what is the volume of B? Give your answer to 3 significant figures. Solution Surface Side Volume Since the volume of A is 8 cm2 Volume The volume of B is the cube is SLIDE 36B The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of A is 8 in2, what is the volume of B? Give your answer to 3 significant figures. SLIDE 37B The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of A is 8 in2, what is the volume of B? Give your answer to 3 significant figures. Solution Surface Side Volume Since the volume of A is 8 in2 Volume The volume of B is the cube is SLIDE 38 What is wrong with this argument? The volume of A is 8 cm3. Therefore, the surface area of A is 24 cm2. SLIDE 39 The volume of A is 8 cm3. Therefore, the surface area of A is 24 cm2. This argument is invalid. We know nothing about the shape of A. If we know that the A is a cube, then this argument becomes valid, because then we can work out that the length of a side of the cube is . Then the surface area is 6 times the surface area of one face: . But we do not know that A is a cube. It could be any shape. For example, there are many cuboids that have a volume of 8 cm3, but different surface areas. SLIDE 38B What is wrong with this argument? The volume of A is 8 ft3. Therefore, the surface area of A is 24 ft2. SLIDE 39B The volume of A is 8 ft3. Therefore, the surface area of A is 24 ft2. This argument is invalid. We know nothing about the shape of A. If we know that the A is a cube, then this argument becomes valid, because then we can work out that the length of a side of the cube is . Then the surface area is 6 times the surface area of one face: . But we do not know that A is a cube. It could be any shape. For example, there are many cuboids that have a volume of 8 ft3, but different surface areas. SLIDE 40 The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of B is 80 cm2, what is the volume of A? Why is the following solution valid when it is not possible to say anything definite about the sides of shapes A and B? Solution Surface �Side� Volume Since the volume of B is 80 cm2 Volume The volume of A is SLIDE 40B The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of B is 80 ft2, what is the volume of A? Why is the following solution valid when it is not possible to say anything definite about the sides of shapes A and B? Solution Surface �Side� Volume Since the volume of B is 80 cm2 Volume The volume of A is SLIDE 41 The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of B is 80 cm2, what is the volume of A? Solution Surface �Side� Volume Since the volume of B is 80 cm2 Volume The volume of A is This solution is valid because it only ever uses ratios. It does not say that the side of A is anything, it only uses the ratio of a possible side of A to that of B. SLIDE 41B The similar shapes, A and B, are such that their surface areas are in the ratio . If the volume of B is 80 ft2, what is the volume of A? Why is the following solution valid when it is not possible to say anything definite about the sides of shapes A and B? Solution Surface �Side� Volume Since the volume of B is 80 cm2 Volume The volume of A is This solution is valid because it only ever uses ratios. It does not say that the side of A is anything, it only uses the ratio of a possible side of A to that of B. SLIDE 42 Without using a calculator to find a cube root, and by means of trial and improvement find to 2 decimal places. You may use a calculator to find cubes. SLIDE 43 too small too big too small too big 1.25 too small 1.26 too big too small SLIDE 44 Evaluate ? ? ? ? ? ? ? ? ? ? SLIDE 45 ? ? ? ? ? ? ? ? ? ?