Introduction to Indices

Hello, everyone! Today we’re going to talk about something exciting in maths: indices (also called powers). An index tells us how many times to multiply a number by itself. For example, in the expression ( 2^3 ), the number 2 is called the base, and 3 is the index. It means ( 2 \times 2 \times 2 ), which equals 8.

What is a Missing Index or Base?

Sometimes, we might have a number with either a missing base or a missing index. Our job is to find out what that missing part is!

Example 1: Finding the Missing Index

Let’s say we have the equation ( 5^x = 125 ). Here, we want to find the missing index ( x ).

  1. Think about what ( 125 ) is. We know that ( 5 \times 5 \times 5 = 125 ).
  2. Now we can see that ( 5^3 = 125 ).
  3. So, we can conclude that ( x = 3 ).

Example 2: Finding the Missing Base

Now, let’s look at a different example: ( x^3 = 64 ). Here, we are trying to find the missing base ( x ).

  1. Think about what ( 64 ) is. We know that ( 4 \times 4 \times 4 = 64 ).
  2. So we can say that ( 4^3 = 64 ).
  3. Thus, we find that ( x = 4 ).

Key Rules for Finding Missing Indices or Bases

  1. Remember the basic powers: Know the small numbers raised to powers (like ( 2^3 = 8, 3^2 = 9 ), etc.).
  2. Use trial and error: If you’re unsure, try different numbers to see if they fit.
  3. Write it down: Sometimes writing out the calculations can help you see the answer more clearly.

Tips and Tricks

  • Use a calculator for larger numbers to check your work.
  • Remember, if the base is negative, make sure to consider whether the index is even or odd.
  • Practice with different numbers to get comfortable with finding missing parts.

Questions to Practice

Easy Level Questions (Find the missing index or base)

  1. ( 3^x = 27 )
  2. ( x^2 = 16 )
  3. ( 2^x = 8 )
  4. ( 5^3 = 125 ) (Find x)
  5. ( x^4 = 81 )
  6. ( 6^x = 36 )
  7. ( x^2 = 25 )
  8. ( 7^x = 49 )
  9. ( 10^x = 100 )
  10. ( x^3 = 1 )
  11. ( 2^5 = 32 ) (Find x)
  12. ( x^2 = 64 )
  13. ( 4^x = 16 )
  14. ( 3^x = 9 )
  15. ( 8^x = 64 )
  16. ( x^2 = 49 )
  17. ( 2^x = 2 )
  18. ( 5^2 = x )
  19. ( x^3 = 8 )
  20. ( 4^x = 64 )

Medium Level Questions

  1. ( 2^x = 16 )
  2. ( x^2 = 121 )
  3. ( 9^x = 27 )
  4. ( 5^x = 625 )
  5. ( x^3 = 729 )
  6. ( 3^x = 81 )
  7. ( 10^x = 1000 )
  8. ( x^2 = 100 )
  9. ( 16^x = 256 )
  10. ( x^4 = 256 )
  11. ( 7^x = 343 )
  12. ( x^3 = 27 )
  13. ( 4^x = 16 )
  14. ( x^2 = 36 )
  15. ( 5^x = 25 )
  16. ( 2^x = 32 )
  17. ( x^2 = 9 )
  18. ( 3^x = 243 )
  19. ( 6^x = 36 )
  20. ( x^3 = 125 )

Hard Level Questions

  1. ( 2^{x+1} = 32 )
  2. ( 5^{2x} = 625 )
  3. ( 3^{x-1} = 27 )
  4. ( 4^{x+2} = 64 )
  5. ( x^{2x} = 16 )
  6. ( 2^{x-3} = 8 )
  7. ( 10^{x-1} = 10 )
  8. ( 2^{2x} = 16 )
  9. ( 7^{x+1} = 343 )
  10. ( x^{x-1} = 27 )
  11. ( 5^{x+2} = 125 )
  12. ( 3^{x+2} = 243 )
  13. ( x^{x} = 64 )
  14. ( 4^x = 1 )
  15. ( 10^{3-x} = 1000 )
  16. ( 6^{2x} = 36 )
  17. ( x^{x+1} = 81 )
  18. ( 2^{x-1} = 4 )
  19. ( 9^{x} = 729 )
  20. ( 8^x = 512 )

Answers

Easy Level Answers

  1. ( 3^3 = 27 ) so ( x = 3 ).
  2. ( 4^2 = 16 ) so ( x = 2 ).
  3. ( 2^3 = 8 ) so ( x = 3 ).
  4. ( 5^3 = 125 ) so ( x = 3 ).
  5. ( 3^4 = 81 ) so ( x = 4 ).
  6. ( 6^2 = 36 ) so ( x = 2 ).
  7. ( 5^2 = 25 ) so ( x = 2 ).
  8. ( 7^2 = 49 ) so ( x = 2 ).
  9. ( 10^2 = 100 ) so ( x = 2 ).
  10. ( 1^3 = 1 ) so ( x = 0 ).
  11. ( 2^5 = 32 ) so ( x = 5 ).
  12. ( 8^2 = 64 ) so ( x = 4 ).
  13. ( 4^2 = 16 ) so ( x = 2 ).
  14. ( 3^2 = 9 ) so ( x = 2 ).
  15. ( 8^2 = 64 ) so ( x = 2 ).
  16. ( 7^2 = 49 ) so ( x = 2 ).
  17. ( 2^1 = 2 ) so ( x = 1 ).
  18. ( 5^2 = 25 ) so ( x = 2 ).
  19. ( 2^3 = 8 ) so ( x = 3 ).
  20. ( 4^3 = 64 ) so ( x = 3 ).

Medium Level Answers

  1. ( 2^4 = 16 ) so ( x = 4 ).
  2. ( 11^2 = 121 ) so ( x = 11 ).
  3. ( 9^{2/3} = 27 ) so ( x = \frac{2}{3} ).
  4. ( 5^4 = 625 ) so ( x = 4 ).
  5. ( 9^3 = 729 ) so ( x = 3 ).
  6. ( 3^4 = 81 ) so ( x = 4 ).
  7. ( 10^3 = 1000 ) so ( x = 3 ).
  8. ( 10^2 = 100 ) so ( x = 2 ).
  9. ( 16^{1/2} = 256 ) so ( x = 2 ).
  10. ( 4^4 = 256 ) so ( x = 4 ).
  11. ( 7^3 = 343 ) so ( x = 3 ).
  12. ( 3^3 = 27 ) so ( x = 3 ).
  13. ( 4^2 = 16 ) so ( x = 2 ).
  14. ( 6^6 = 36 ) so ( x = 6 ).
  15. ( 5^2 = 25 ) so ( x = 2 ).
  16. ( 2^5 = 32 ) so ( x = 5 ).
  17. ( 3^2 = 9 ) so ( x = 3 ).
  18. ( 3^4 = 81 ) so ( x = 4 ).
  19. ( 6^2 = 36 ) so ( x = 2 ).
  20. ( 5^3 = 125 ) so ( x = 3 ).

Hard Level Answers

  1. ( 2^{x+1} = 32 ) so ( x = 4 ).
  2. ( 5^{2x} = 625 ) so ( x = 2 ).
  3. ( 3^{x-1} = 27 ) so ( x = 4 ).
  4. ( 4^{x+2} = 64 ) so ( x = 2 ).
  5. ( x^{2x} = 16 ) so ( x = 2 ).
  6. ( 2^{x-3} = 8 ) so ( x = 6 ).
  7. ( 10^{x-1} = 10 ) so ( x = 2 ).
  8. ( 2^{2x} = 16 ) so ( x = 2 ).
  9. ( 7^{x+1} = 343 ) so ( x = 2 ).
  10. ( x^{x-1} = 27 ) so ( x = 3 ).
  11. ( 5^{x+2} = 125 ) so ( x = 1 ).
  12. ( 3^{x+2} = 243 ) so ( x = 3 ).
  13. ( x^{x} = 64 ) so ( x = 4 ).
  14. ( 4^0 = 1 ) so ( x = 0 ).
  15. ( 10^{3-x} = 1000 ) so ( x = 0 ).
  16. ( 6^{2x} = 36 ) so ( x = 1 ).
  17. ( x^{x+1} = 81 ) so ( x = 3 ).
  18. ( 2^{x-1} = 4 ) so ( x = 3 ).
  19. ( 9^{x} = 729 ) so ( x = 3 ).
  20. ( 8^x = 512 ) so ( x = 3 ).

That’s it for today! Remember to practice, and don’t hesitate to ask questions if you’re unsure about anything. Happy studying!