The laws of indices - negative and zero exponents | Maths

Starter quiz

What is the value of $x$ for $(m^x)^8 = m^{24}$

'3' ✓
What is the value of $x$ for $(m^5)^5 = m^x$

'25' ✓
What is the value of $x$ for $(m^3)^x = m^9$

'3' ✓
What is the value of $x$ for $(m^8)^3 = m^x$

'24' ✓
What is the value of $x$ for $\frac{1}{4} = \frac{1}{2^x}$

'2' ✓
What is the value of $x$ for $\frac{1}{64} = \frac{1}{2^x}$

'6' ✓

Exit quiz

What is the value of $x$ for $(m^x)^{-8} = m^{24}$

'-3' ✓
What is the value of $x$ for $(m^{-5})^5 = m^x$

'-25' ✓
What is the value of $x$ for $(m^3)^x = m^{-3}$

'-1' ✓
What is the value of $x$ for $\frac{1}{4} = 2^x$

'-2' ✓
What is the value of $x$ for $\frac{1}{64} = 2^x$

'-6' ✓
What is the value of $a$ for $5^8 \times 5^a = 5^6$ ?

'-2' ✓

Worksheet

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Lesson Details

Key learning points

A calculator can be helpful to explore what happens when the exponent is zero or negative.
Any term with an exponent of zero is equal to one.
A negative exponent tells you how many times you need to multiply the reciprocal of the base.
Negative exponents appear when writing very small numbers in standard form.

Common misconception

Any number with an index of 0 equates to 0

Reiterate the division of any number by itself is 1. This applies with laws of indices too. E.g 8^4 / 8^4 = 8^0. Any number divided by itself gives 1, therefore 8^0 = 1. For more advanced understanding, algebraic can be used instead of numbers.

Keywords

Reciprocal - A reciprocal is the multiplicative inverse of any non-zero number. Any non-zero number multiplied by its reciprocal is equal to 1
Indices - An exponent is a number positioned above and to the right of a base value. It indicates repeated multiplication. An alternative word for this is index (plural indices).