Simplify and evaluate powers lesson | Mathematical Methods (Year 11, VCE Units 1 & 2)

Index laws

Index laws simplify powers with compatible bases. The operation between the powers determines whether exponents are added, subtracted or multiplied

a^{m}\times a^{n}=a^{m+n}\quad a^{m}\div a^{n}=a^{m-n}\quad(a\ne 0)\quad \left(a^{m}\right)^{n}=a^{mn}

Multiply equal bases When equal bases are multiplied, add their exponents \(\displaystyle a^{m}\times a^{n}=a^{m+n}\)
Divide equal bases When equal bases are divided, subtract the denominator exponent from the numerator exponent \(\displaystyle a^{m}\div a^{n}=a^{m-n}\quad(a\ne 0)\)
Raise a power to a power When a power is raised to another power, multiply the exponents \(\displaystyle \left(a^{m}\right)^{n}=a^{m\times n}\)
Interpret negative and rational indices A negative index gives a reciprocal, and a rational index gives a root followed by an outside power \(\displaystyle a^{-m}=\frac{1}{a^{m}}\quad(a\ne 0),\quad a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^{m}\)

Simplify and evaluate the expression below

\left(3^{2}\times 3^{-1}\right)^{3}

\(\displaystyle 3^{2}\times 3^{-1}=3^{2+(-1)}=3^{1}=3\)The equal bases are multiplied, so their exponents are added
\(\displaystyle \left(3^{1}\right)^{3}=3^{1\times 3}=3^{3}\)A power raised to a power uses the product of the exponents
\(\displaystyle 3^{3}=3\times 3\times 3=27\)Evaluate the remaining positive integer power to obtain 27

Adding exponents for a power of a power

✕\(\displaystyle \left(2^{3}\right)^{2}=2^{3+2}=2^{5}=32\)✓\(\displaystyle \left(2^{3}\right)^{2}=2^{3\times 2}=2^{6}=64\)

A power raised to a power multiplies exponents; adding exponents is used for multiplying equal bases

Multiplying exponents in a product

✕\(\displaystyle 2^{3}\times 2^{2}=2^{3\times 2}=2^{6}\)✓\(\displaystyle 2^{3}\times 2^{2}=2^{3+2}=2^{5}=32\)

Equal bases in a product require the exponents to be added, not multiplied

Applying a law across a sum

✕\(\displaystyle 2^{3}+2^{2}=2^{3+2}=32\)✓\(\displaystyle 2^{3}+2^{2}=8+4=12\)

Index laws combine equal bases in products or quotients, not terms joined by addition

introOne index law is used with integer exponents

standardA product and quotient are combined, including negative input exponents and positive final indices

challengeFractional indices and nested powers require exact root or power reasoning

examA compact formula choice or coefficient and exponent result tests several index laws

Why are exponents added when equal bases are multiplied?+

When powers with the same base are multiplied, their exponents are added because their factors are combined

What does a negative exponent mean?+

For a non-zero base, a negative exponent gives the reciprocal of the corresponding positive power

How is a rational exponent evaluated?+

For a positive base, take the positive root specified by the denominator, then raise it to the power specified by the numerator

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 11: Exponentials and logarithms