Top Rated by Parents/Students Nationwide

07478355331

Standard Form

Standard Form

Converting To & From Standard Form

What is standard form?

  • Standard Form (sometimes called Standard Index Form) is a way of writing very big and very small numbers using powers of 10

Why do we use standard form?

  • Writing big (and small) numbers in Standard Form allows us to:
    • write them more neatly
    • compare them more easily
    • and it makes things easier when doing calculations

How do we use standard form?

  • Using Standard Form numbers are always written in the form:

  • The rules:
    • 1 ≤ a < 10 so there is one non-zero digit before the decimal point
    • n > 0 for LARGE numbers – how many times a is multiplied by 10
    • n < 0 for SMALL numbers – how many times a is divided by 10
    • Do calculations on a calculator (if allowed)

Worked example

Worked example

Without a calculator, write 0.007052  in standard form.

 

Standard form will be written as a × 10n. Ignore the place value and find the leading non-zero digit. Use this to find the value of a.

a = 7.052

The original number is smaller than 1 so n will be negative. Count how many times you need to divide a by 10 to get the original number.

0.007052 = 7.052 ÷ 10 ÷ 10 ÷ 10

Therefore n = -3.

0.007052 = 7.052 × 10-3 

Operations with Standard Form

How do I multiply or divide two numbers in standard form?

  • If you can, use a calculator!
  • Otherwise multiply/divide the number parts first
    • If this answer is less than 1 or 10 or more then you will need to write it in standard form again
      • e.g. 4 × 5 = 20 = 2 × 10 1 or 2 ÷ 4 = 0.5 = 5 × 10 -1
  • Then multiply/divide the powers of 10 using the laws of indices
  • Multiply the two parts together to get your answer in standard form
    • You might have to use the laws of indices one more
      • e.g. 4 × 10 2 × 5 × 10 7= 2 × 10 1 × 10 9 = 2 × 10 10

How do I add or subtract two numbers in standard form?

  • If you can, use a calculator!
  • If the two numbers have the same power of 10 then you can simply add/subtract the number parts
    • If the answer is less than 1 or 10 or more then you will have to rewrite in standard form
      • e.g. 7 × 10 5 – 6.2 × 10 5 =  0.8 × 10 5 = 8 × 10 -1 × 10 5 = 8 × 10 4 
  • Otherwise convert both numbers so that they have the same power of 10 (choosing the larger power)
    • e.g. 7 × 10 5 + 6 × 10 4 = 7 × 10 5 + 0.6 × 10 5 = 7.6 × 10 5
  • If the powers of 10 are small then you might find it easier to convert both numbers to ordinary numbers before adding/subtracting
    • You can convert your answer back to standard form if needed

How do I find powers or roots of a number in standard form?

  • If you can, use a calculator!
  • As standard form is two terms multiplied together you can split the power or root up
  • Check to see whether you have to write your final answer in standard form

Worked example

(a)
Without using a calculator, multiply by  .
Give your answer in standard form.

Separate into numbers and powers of 10.

Multiply the integers together.
Use the laws of indices on the powers of 10.

Adjust the first number, , such that 

Write in standard form.

(b)
Use your calculator to find  .
Write your answer in the form  , where  and  is an integer.

Input the calculation into your calculator.
The result may or may not be in standard form.
Copy the digits, especially those zeros, carefully!

Re-write in standard form.

Number Toolkit
Prime Factors, HCF & LCM
Powers, Roots & Standard Form
Fractions
Percentages
Simple & Compound Interest, Growth & Decay
Fractions, Decimals & Percentages
Rounding, Estimation & Bounds
Surds
Using a Calculator
Algebra Toolkit
Algebraic Roots & Indices
Expanding Brackets
Factorising
Completing the Square
Rearranging Formulae
Algebraic Proof
Linear Equations
Solving Quadratic Equations
Simultaneous Equationsr
Iteration
Forming & Solving Equations
Functions
Coordinate Geometrys
Linear Graphs y = mx + c
Graphs of Functions
Equation of a Circle
Estimating Gradients & Areas under Graphs
Real-Life Graphs
Solving Inequalities
Graphing Inequalities
Transformations of Graphs
Sequences
Ratio Toolkit
Ratio Problem Solving
Direct & Inverse Proportions
Standard & Compound Units
Exchange Rates & Best Buys
Geometry Toolkit
Angles in Polygons & Parallel Lines
Bearings, Scale Drawing, Constructions & Loci
Circle Theorems
Area & Perimeter
Circles, Arcs & Sectors
Volume & Surface Area
Congruence, Similarity & Geometrical Proof
Area & Volume of Similar Shapes
Right-Angled Triangles – Pythagoras & Trigonometry
Sine, Cosine Rule & Area of Triangles
3D Pythagoras & Trigonometry
Vectors
Transformations
Probability Toolkit
Simple Probability Diagrams
Tree Diagrams
Combined & Conditional Probability
Statistics Toolkit
Statistical Diagrams
Histograms
Cumulative Frequency & Box Plots
Scatter Graphs & Correlation

Leading tutoring agency within the North East, that specialises in getting students top grades all over the country.

Quick Links

Social

Get In Touch

Get Started

Book A Demo

  • 2024 League Of Learning