Indices and Standard Form: notes

Laws of Indices

• Multiply: a^m x a^n = a^(m+n). Add the powers when the bases are the same.
• Divide: a^m / a^n = a^(m-n). Subtract the powers.
• Power of a power: (a^m)^n = a^(mn). Multiply the powers.
• Zero power: a^0 = 1 for any a that is not 0.
• Negative power: a^(-n) = 1 / a^n. So 2^(-3) = 1/8.
• Fractional power: a^(1/2) is the square root of a. a^(1/3) is the cube root. 8^(2/3) = (cube root of 8)^2 = 4.

Standard Form

• Standard form: a x 10^n where a is at least 1 and less than 10 and n is a whole number.
• Large numbers: 45 000 = 4.5 x 10^4. Move the point left; n is positive.
• Small numbers: 0.0032 = 3.2 x 10^(-3). Move the point right; n is negative.
• Multiplying: (2 x 10^3) x (3 x 10^4) = 6 x 10^7. Multiply the numbers, add the powers.
• Adding: make the powers the same first. Then add the numbers and write the answer back in standard form.

Index Equations

• Same base: if 2^x = 32, write 32 as 2^5, so x = 5.