Indices and Standard Form

In this topic you will learn how to work with powers (indices) and how to write very large or very small numbers using standard form. We cover the rules for positive, zero, negative and fractional indices, and how to simplify expressions quickly.

  • Multiplying and dividing powers
  • Zero and negative indices
  • Fractional indices (roots)
  • Converting to and from standard form

Lessons for Indices and Standard Form

Positive Indices and the Basic Rules

What is an index?The index (or power/exponent) tells us how many times to multiply the base by itself.Example: 53 = …

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Zero and Negative Indices

Zero IndexAny non-zero number to the power 0 equals 1.a0 = 1 (where a ≠ 0)Example: 70 = 1, (3x)0 …

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Fractional Indices (Roots)

Fractional Indicesa1/n = n√a (the nth root of a)am/n = (n√a)m = n√amExamples91/2 = √9 = 381/3 = 3√8 = …

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Standard Form

What is Standard Form?A way of writing very large or very small numbers as a × 10n where 1 ≤ …

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Exercises for Indices and Standard Form

Multiply and Divide Powers

Simplify fully:x4 × x3a9 ÷ a2(m2)5p6 × p × p2(23)2

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Simplify Negative Powers

Write with positive indices only:x-35-21/y-52a-1(3x)-2

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Evaluate Roots

Evaluate:161/2641/3253/282/3813/4

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Convert to Standard Form

Write in standard form:780000.0005642.10.097391000000

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Mixed Practice

Simplify:(x3 × x4) ÷ x2(a2)3 × a532 × 34(y5 ÷ y2)2

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Evaluate

Evaluate without a calculator:40 + 3-110-22-3 × 25(50)-3

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Simplify Algebraic Expressions

Simplify, leaving answers with positive indices:x1/2 × x1/2(y1/3)6a2/3 ÷ a1/3(8x3y6)1/3

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Calculate in Standard Form

Calculate, giving answers in standard form:(3 × 105) × (2 × 103)(6 × 107) ÷ (3 × 104)(4 × 10-2) …

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