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Writing indices and index laws

Writing indices and index laws

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Writing indices and index laws

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Explainer Video

Tutor: Bilal

Summary

Writing indices and index laws

​​In a nutshell

Indices are used as shorthand for repeated multiplication. Another term for indices is "powers".



Writing indices

Indices are written as a small number to the top right of another number. The index number (also called the power) tells you the number of times you multiply the bigger number by itself.


Examples
  • 545^454​ is shorthand for 5×5×5×5=6255\times5\times5\times5=6255×5×5×5=625. It is read as "five to the power of four".
  • ​15215^2152​ is shorthand for 15×15=22515\times15=22515×15=225. Anything to the power of two is called "squared". In this case, this is read as "fifteen squared".
  • ​131^313​ is shorthand for 1×1×1=11\times1\times1=11×1×1=1. Anything to the power of three is called "cubed". In this case, this is read as "one cubed".



Index laws

Index laws (or the laws of indices) are a set of rules that help with writing powers or multiplying/dividing with them. Here are the rules:


The six index laws

  1. When multiplying two powers of the same number, add the powers.
  2. When dividing two powers of the same number, subtract the powers.
  3. When raising a power to another power, multiply the powers.
  4. A negative power is the same as turning the number "upside down" like a fraction.
  5. ​Anything to the power of 111 is itself.
  6. Anything to the power of 000 is 111.​


Examples
  1. ​63×65=63+5=686^3\times6^5=6^{3+5}=6^863×65=63+5=68​​
  2. ​137÷13−2=137−(−2)=13913^7\div13^{-2}=13^{7-(-2)}=13^9137÷13−2=137−(−2)=139​​
  3. ​(28)3=28×3=224(2^8)^3=2^{8\times3}=2^{24}(28)3=28×3=224​​
  4. ​3−5=1353^{-5}=\frac{1}{3^5}3−5=351​​​
  5. ​331=3333^1=33331=33​​
  6. ​330=133^0=1330=1​​


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FAQs - Frequently Asked Questions

What is anything to the power of 0?

Anything to the power of 0 is 1.

What is anything to the power of 1?

Anything to the power of 1 is itself.

How do you raise one power to another?

When raising one power to another, multiply the powers.

How do you divide two powers?

When dividing two powers of the same number, subtract the powers.

How do you multiply two powers together?

When multiplying two powers of the same number, add the powers.

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