Dr. J's Maths.com

**Where the techniques of Maths**

are explained in simple terms.

are explained in simple terms.

Algebra - Indices and Powers - basic questions.

Test yourself 1.

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Simplify the following expressions to evaluate your understanding of the five basic concepts underlying this topic.

Use the HINTS given if you need to.

TECHNIQUE: Be systematic in your solving of mathematics problems. Many students just begin anywhere and then wonder why they obtain incorrect answers.

In questions such as those below, move left to right by viewing the same types of terms. Often that requires you to collect the numbers first and then the first pronumeral terms together, then the second pronumerals, etc.

In some questions you need to simplify brackets - and comments to help you are made here or on the Solutions page.

Multiplication and division of powers:

1. Simplify: 3a × 2b × 5c × 4d

Answer.120abcd.2. Simplify: y ^{3}× y^{5}

Answer.y^{8}.3. Simplify: 6 ^{2}× 6^{-5}

Answer.6^{-3}= 1/(6^{3}) = 1/216.4. Simplify: 2k ^{3}× 3k^{4}.

Answer.6k^{7}.5. Simplify: 3x ^{2}y × 5xy^{3}

Answer.15x^{3}y^{4}.6. Simplify 7 ^{ -2}× 7^{5}× 7^{ -2}

Answer.7.7. Simplify: .

Answer.3m/2.8.Simplify: .

Answer.(4a^{2})/(3b^{4}).9. Simplify: 10x ^{10}÷ 5x^{6}

Answer.2x^{4}.10. Simplify: .

Answer.(13y^{3}) / (x7^{12}).11. Simplify:

Answer.(9y^{4}) / (4x^{10}).12. Simplify: 3j

Hint.Start by multiplying the constant terms and record your answer. Then combine the j terms, then the k terms and so on.^{2}× 2k^{3}m^{2}× 3jkmn^{2}

Answer.18j^{3}k^{4}m^{3}n^{2}.Powers of powers:

13. Simplify: (k ^{3})^{5}

Answer.k^{15}.14. Simplify: (c ^{-3})^{-2}

Answer.c^{6}.15. Simplify: (x ^{-½})^{-2}

Answer.x.16. Simplify fully: ((d ^{2})^{-3})^{-1}

Hint.Start by multiplying the inner exponents (2 × -3). Then multiply by the -1. So you move from inside to outside.

Answer.d^{6}.17. Simplify fully: (3x

Hint.Expand the brackets by applying the outside exponent to each term in order.^{2}y^{3}z)^{4}

Answer.81x^{8}y^{12}z^{4}.18. Simplify fully: (2m

^{3})^{3}× 4m^{5}

Answer.32^{4}.

Negative powers:

19. Rewrite with a negative index.

Answer.-2/x.20. Rewrite -2x ^{-1}with a positive index.

Answer.-2/x.21. Simplify: (3 ^{-1})^{2}× 27

Answer.3.22. Simplify . 23. Remove the brackets: (3m)

^{-3}

Answer.1/27m^{3}.24. Simplify: .

Hint.Start by removing the negative sign from outside the brackets and simply turn the inside fraction upside down.

Exponents of 1 or zero:

25. Simplify: 5 ^{0}+ 101^{0}

Answer.Sum = 2.26. Simplify: -(4m) ^{0}

Answer.-1.27. Simplify: - 7b ^{0}

Answer.-7.28. Simplify: (3 ^{3})^{0}× 5^{2}

Answer.25.29. Simplify: 8m ^{0}÷ (2m)^{0}

Hint.The first exponent of zero only applies to the m term.

Answer.8.30. Simplify: (2x ^{3 }- 4y^{2})^{1}

Answer.2x^{3 }- 4y^{2}.

Fractional indices:

31. Write with a fractional index: .

32. Evaluate

Answer.6.33. Write with positive indices 34. Rewrite and simplify using a fractional index:

Hint.Rewrite the square root as x to the half then add the exponents.

35. Write in a format with radical signs: . 36. Write in a different format: .

Mixtures:

37. Simplify: (3t) ^{2}× 4t^{3}

Answer.36t^{5}.38. Simplify: .

Answer.49.39. Simplify:

Answer.3^{23}.40. Simplify: 15m

Answer.3m^{4}n^{5}÷ 5m^{2}n^{4}^{2}n.41. Simplify: (3g

Hint.With this question and the next, expand the brackets first by multiplying the exponents then change to negative indices for the divisions. Remember to keep working left to right in a methodological way and do not try to do too much at once.^{3})^{3}÷ (g^{2})^{4}

Answer.27g.42. Simplify: (10x

Answer.250x^{2}y^{-2})^{3}÷ (2x^{-1}y^{3})^{2}^{8}.