Integer Powers and Associated Real Roots

What Are Indices?

Indices, or exponents, are small numbers positioned slightly above and to the right of regular numbers, like in . The index in this case is the ‘3’.

Indices indicate that a number should be multiplied by itself a certain number of times. For example, you’ll see indices in square numbers, cube numbers and even higher powers.

Square Numbers

Square numbers are the result of multiplying a whole number by itself. They are a special category of numbers with unique properties.

Let’s look at the first five square numbers:

• The first square number is 1, which is

• The second square number is 4, calculated as

• The third square number is 9, calculated as

• The fourth square number is 16, calculated as

• The fifth square number is 25, calculated as

Each of these can be represented as , where n is a whole number. For instance, equals 16, which is mathematically expressed as and read as “four squared.”

Cube Numbers

In a similar way, cube numbers are the result of multiplying a number by itself and then multiplying that product by the number once more.

In other words, cubed numbers are formed by multiplying a number by itself two additional times.

Let’s look at the first five cubed numbers:

• The first cube number is 1, calculated as

• The second cube number is 8, calculated as

• The third cube number is 27, calculated as

• The fourth cube number is 64, calculated as

• The fifth cube number is 125, calculated as

Cubed numbers are often used to calculate the volume of cubes in geometry.

The Concept of ‘Power’

In mathematical expressions like or , the small number on the right (the superscript) is known as the “exponent” or “power.” It signifies the number of times the base (the larger number) should be multiplied by itself.

The concept of power is a fundamental part of arithmetic and algebra, and it extends beyond squares and cubes. For example, would be .

Raising a number to a power is not limited to whole numbers. It can also be applied to fractions and decimals, although the calculations can become more complex.

From Powers to Roots

Just as numbers can be raised to a power to generate square or cubed numbers, the reverse process also exists. This operation is known as finding the “root” of a number.

Essentially, while powers multiply the base number by itself a certain number of times, roots identify which number was raised to a certain power to obtain a given number.

There are different types of roots, but square and cube roots are the most commonly used.

Square Roots

The square root of a number is essentially the reverse operation of squaring that number and is denoted by the symbol √. In other words, if , then .

For example, the square root of 16 is 4, because . Similarly, the square root of 25 is 5, as .

Let’s use the numbers 3 and 9 to look at how squares and square roots are linked.

1. Starting with 3, if you square this number, you get 9 .

2. If you then take the square root of 9, you return to the original number, 3 .

• Squaring a number (like 3) gives you a square number (like 9)

• The square root operation then reverses this, taking the square number (9) back to its original form (3)

The square root “undoes” the operation of squaring, and vice versa. So, squaring and then taking the square root of a number will give you back the original number.

Cube Roots

Just like the square root, the cube root of a number is a value that, when cubed, will produce the original number. Cube roots are generally represented by the symbol ∛.

For example, the cube root of 27 is 3 because . Similarly, the cube root of 64 is 4 because .

Practice Questions

Question 1:

What is

Solution

means , which equals 9.

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Question 2:

What is the square root of 81?

Solution

The square root of 81 is 9, as .

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Question 3:

What is

Solution

signifies , which results in 64.

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Question 4:

What is the cube root of 125?

Solution

The cube root of 125 is 5, as .