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Powers and roots: Square and cube numbers

Powers and roots: Square and cube numbers

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OCRPearson EdexcelAQA

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Summary

Powers and roots: Square and cube numbers

​​In a nutshell

A square number is the name given to the answer when a number is multiplied by itself. A cube number is the name given when a number is multiplied by itself twice. A square root is the inverse of a square, and a cube root is the inverse of a cube. When a number is multiplied by itself more than twice, it can written using powers.



Powers

When a number is multiplied by itself several times, it can be written using powers.

​ama^m​​


Where aa is referred to as the base, and mm is the power - also known as the index or exponent.


Example 1

What is 242^4?


​22 is multiplied by itself 44 times.​

​24=2×2×2×2=16‾2^4 = 2 \times 2 \times 2 \times 2 = \underline{16}​​


​

Square numbers

A square number is the product when a number is multiplied by itself. You have to know the first 1515 square numbers.​

12=1×1=1‾22=2×2=4‾32=3×3=9‾42=4×4=16‾52=5×5=25‾62=6×6=36‾72=7×7=49‾82=8×8=64‾92=9×9=81‾102=10×10=100‾112=11×11=121‾122=12×12=144‾132=13×13=169‾142=14×14=196‾152=15×15=225‾1^2 =1 \times 1 = \underline1 \\2^2 =2 \times 2 = \underline4 \\3^2 =3 \times 3 = \underline9 \\4^2 =4 \times 4 = \underline{16} \\5^2 =5 \times 5 = \underline{25} \\6^2 =6 \times 6 = \underline{36} \\7^2 =7 \times 7 = \underline{49} \\8^2 =8 \times 8 = \underline{64} \\9^2 =9 \times 9 = \underline{81} \\10^2 =10 \times 10 = \underline{100} \\11^2 =11 \times 11 = \underline{121} \\12^2 =12 \times 12 = \underline{144} \\13^2 =13 \times 13 = \underline{169} \\14^2 =14 \times 14 = \underline{196} \\15^2 =15 \times 15 = \underline{225} \\​​



Cube numbers

A cube number is the product when a number is multiplied by itself twice. You have to know the first 55 cube numbers.

13=1×1×1=1‾23=2×2×2=8‾33=3×3×3=27‾43=4×4×4=64‾53=5×5×5=125‾1^3 = 1 \times 1 \times 1 = \underline1 \\2^3 = 2 \times 2 \times 2 = \underline8 \\3^3 = 3 \times 3 \times 3 = \underline{27} \\4^3 = 4 \times 4 \times 4 = \underline{64} \\5^3 = 5 \times 5 \times 5 = \underline{125} \\​​



Square and cube roots

To find a square root, think of which number you would need to multiply itself by to get a square number. To find a cube root, think of which number was multiplied by itself twice.


​​Example 2

What is 81\sqrt {81} ?

​

Find which positive number - when multiplied by itself - gives 8181.

9×9=819\times 9 = 81​​

​

​81=9‾\underline{\sqrt{81} = 9}

​

Note: Square roots are defined to only be positive.


Example 3

Find the value of 1253\sqrt[3]{125}.


Find the number that - when multiplied by itself twice - gives 125125.

5×5×5=1255 \times 5 \times 5 = 125

​​

​1253=5‾\underline{\sqrt[3]{125}=5}


​

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Learn with Basics

Length:
Standard form

Unit 1

Standard form

Writing indices and index laws

Unit 2

Writing indices and index laws

Jump Ahead

Powers and roots: Square and cube numbers

Unit 3

Powers and roots: Square and cube numbers

Final Test

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FAQs - Frequently Asked Questions

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