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Welcome to GCSE Edexcel Maths revision.

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Topic M 4: Powers, roots and standard form.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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In 3 to the power of open bracket 4 close bracket , 3 is the base and 4 is the index: 3 multiplied by 3 multiplied by 3 multiplied by 3 equals 81.

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Squaring and cubing mean powers 2 and 3.

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A power is not base multiplied by index.

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Square roots undo squaring: square root of 49 equals 7.

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The symbol square root of gives the non-negative square root; the equation x squared equals 49 has two solutions, x equals plus or minus 7.

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Cube roots can be negative: cube root of open bracket minus 8 close bracket equals minus 2.

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Recognise useful powers: 2 to the power of open bracket 5 close bracket equals 32, 3 cubed equals 27, 4 cubed equals 64 and 5 cubed equals 125.

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Know common squares and cubes so you can undo familiar powers without a calculator.

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With the same base,

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multiply by adding indices and divide by subtracting: a to the power of open bracket m close bracket a to the power of open bracket n close bracket equals a to the power of open bracket m plus n close bracket ;

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the fraction with numerator open bracket a to the power of open bracket m close bracket close bracket and denominator open bracket a to the power of open bracket n close bracket close bracket ,

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end fraction equals a to the power of open bracket m minus n close bracket for a is not equal to 0.

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A power of a power multiplies indices: open bracket a to the power of open bracket m close bracket close bracket to the power of open bracket n close bracket equals a to the power of open bracket m n close bracket .

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Also open bracket a b close bracket to the power of open bracket n close bracket equals a to the power of open bracket n close bracket b to the power of open bracket n close bracket .

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You cannot apply these rules to sums: open bracket a plus b close bracket squared is not a squared plus b squared .

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For a non-zero base,

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a to the power of open bracket 0 close bracket equals 1 and a to the power of open bracket minus n close bracket equals the fraction with numerator open bracket 1 close bracket and denominator open bracket a to the power of open bracket n close bracket close bracket ,

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end fraction .

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For example, 2 to the power of open bracket minus 3 close bracket equals 1 over 8 .

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A negative index does not make the value negative.

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Standard form is A multiplied by 10 to the power of open bracket n close bracket ,

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where 1 is less than or equal to A is less than 10 and n is an integer. 43,000 equals 4.3 multiplied by 10 to the power of open bracket 4 close bracket ;

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0.00072 equals 7.2 multiplied by 10 to the power of open bracket minus 4 close bracket .

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Positive powers of 10 describe large numbers;

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negative powers describe small positive numbers. 0.003 is 3 multiplied by 10 to the power of open bracket minus 3 close bracket ,

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not minus 3000.

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Worked example: open bracket 3 multiplied by 10 to the power of open bracket 5 close bracket close bracket multiplied by open bracket 4 multiplied by 10 to the power of open bracket minus 2 close bracket close bracket equals 12 multiplied by 10 cubed equals 1.2 multiplied by 10 to the power of open bracket 4 close bracket .

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Multiply coefficients, add indices, then adjust the coefficient to the required range.

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Worked example: the fraction with numerator open bracket 6 multiplied by 10 to the power of open bracket 7 close bracket close bracket and denominator open bracket 2 multiplied by 10 cubed close bracket ,

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end fraction equals 3 multiplied by 10 to the power of open bracket 4 close bracket .

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Divide coefficients and subtract indices.

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For addition and subtraction,

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rewrite with the same power of 10: 3.2 multiplied by 10 to the power of open bracket 4 close bracket plus 6 multiplied by 10 cubed equals open bracket 3.2 plus 0.6 close bracket multiplied by 10 to the power of open bracket 4 close bracket equals 3.8 multiplied by 10 to the power of open bracket 4 close bracket .

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On a calculator, use its standard-form entry key and brackets around complete numbers when dividing.

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Check whether the displayed exponent is positive or negative.

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Estimate roots using nearby known powers: since 8 squared is less than 70 is less than 9 squared , square root of 70 lies between 8 and 9.

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Trial values can refine an estimate without implying the root is rational.

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For a positive base,

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a to the power of open bracket the fraction with numerator open bracket 1 close bracket and denominator open bracket n close bracket ,

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end fraction close bracket means the nth root of a,

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and a to the power of open bracket the fraction with numerator open bracket m close bracket and denominator open bracket n close bracket ,

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end fraction close bracket means raise the nth root to power m.

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Thus 27 to the power of open bracket 2 over 3 close bracket equals open bracket cube root of 27 close bracket squared equals 9.

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Worked example: 16 to the power of open bracket the fraction with numerator open bracket minus 3 close bracket and denominator open bracket 4 close bracket ,

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end fraction close bracket equals the fraction with numerator open bracket 1 close bracket and denominator open bracket open bracket fourth root of 16 close bracket cubed close bracket ,

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end fraction equals the fraction with numerator open bracket 1 close bracket and denominator open bracket 2 cubed close bracket ,

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end fraction equals 1 over 8 .

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Handle the negative sign in the index by taking a reciprocal.

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Fractional and negative indices obey the same index laws within their real-number domains.

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For example, x to the power of open bracket 1 over 2 close bracket multiplied by x to the power of open bracket 3 over 2 close bracket equals x squared for x is greater than 0.

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That completes Powers, roots and standard form.

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Revisit the notes and test yourself on the revision website.