An index (or power) is a small number placed in the upper right hand corner of another number to note how many times a base is being multiplied by itself.

10 to the power of 3 is equals to 10 times 10 times 10. 10 is the base, 3 is the exponent or power.

For example, in the expression {10}^3 the number 10 is the base term and the number 3 is the index (or power) term. The expression {10}^3 is the same as 10 \times 10 \times 10, or the number 10 multiplied 3 times.

We often encounter a power of 2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number, e.g. 5 with an exponent (or power) of 2, can be expressed as 5^2, and can be read as "5 to the power of 2" or "five squared".

A number, e.g. 10 to the power of 3, can be expressed as 10^3, and can be read as "ten cubed". A power of 3 is involved in calculations like measuring the volume of a cube.

x squared illustrated as a square. x cubed illustrated as a cube. Ask your teacher for more information.

A base to the power of any other number, e.g. 3^{4}, can be read as "three to the power of four", and means that the base number is multiplied by itself the number of times shown in the power.

\displaystyle 3^{4}

\displaystyle =

\displaystyle 3\times3\times3\times3

Exploration

The following demonstration illustrates more of this notation. Try varying the bases and exponents (by moving the sliders) to see how the numbers change.

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The index is equal to the number of times the base is multiplied by itself.

Examples

Example 1

State the base for the expression 3^{2}.

Worked Solution

Create a strategy

Use the base and power definition: \text{base}^{\text{power}}

Apply the idea

\displaystyle \text{base}^{\text{power}}

\displaystyle =

\displaystyle 3^{2}

The base of the expression is 3.

Example 2

Identify the power for the expression 4^{6}.

Worked Solution

Create a strategy

Use the base and power definition: \text{base}^{\text{power}}

Apply the idea

\displaystyle \text{base}^\text{power}

\displaystyle =

\displaystyle 4^{6}

The power of the expression is 6.

Example 3

Evaluate 3^{5} \div 3^{3}.

Worked Solution

Create a strategy

Write the power terms in expanded notation and then perform the division.

Apply the idea

\displaystyle 3^{5} \div 3^{3}	\displaystyle =	\displaystyle (3 \times 3 \times 3 \times 3 \times 3 ) \div (3 \times 3 \times 3)	Expand the notations
	\displaystyle =	\displaystyle \dfrac{3 \times 3 \times 3 \times 3 \times 3 }{3 \times 3 \times 3}	Write as a fraction
	\displaystyle =	\displaystyle \dfrac{3 \times 3 }{1}	Cancel out 3 \times 3 \times 3
	\displaystyle =	\displaystyle 9	Evaluate

Idea summary

An index (or power) notes how many times a base is being multiplied by itself.

A base to the power of any other number means that the base number is multiplied by itself the number of times shown in the power.

Outcomes

ACMNA149

Investigate index notation and represent whole numbers as products of powers of prime numbers

ACMNA150

Investigate and use square roots of perfect square numbers

1.09 Powers of whole numbers

Ideas

Powers of whole numbers

Exploration

Examples

Example 1

Example 2

Example 3

Outcomes

ACMNA149

ACMNA150

What is Mathspace

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