Indices, roots, standard form and exact calculation Revision Notes

What is Index Notation?

Index notation is simply a shorter way of writing repeated multiplication. Instead of writing 7 × 7 × 7, we write 7³. In this expression, '7' is the base, and '3' is the index (or power or exponent). The index tells you how many times to multiply the base by itself. So, 7³ means '7 multiplied by itself 3 times'.

Another example is 5⁴, where 5 is the base and 4 is the index. This means 5 × 5 × 5 × 5. Calculating this gives 25 × 5 × 5 = 125 × 5 = 625. It's important not to confuse 5⁴ with 5 × 4. The index indicates repeated multiplication, not simple multiplication of base by index.

Parts of an Indexed Expression

In an expression like aⁿ:

• 'a' is the base. This is the number or variable that is being multiplied repeatedly.
• 'n' is the index (also known as the exponent or power). This is the small number written above and to the right of the base. It tells you how many times the base is used as a factor in the multiplication.

For example, in 6², '6' is the base, and '2' is the index. This represents 6 × 6 = 36. If you have x⁵, 'x' is the base, and '5' is the index, meaning x × x × x × x × x. Pay close attention to negative signs. For instance, (-3)² means (-3) × (-3) = 9, while -3² means -(3 × 3) = -9. The base is what is directly affected by the index.

Calculating Values from Index Notation

To calculate the value of a number written in index notation, you perform the repeated multiplication indicated by the index.

Examples:

• 4³ = 4 × 4 × 4 = 16 × 4 = 64.
• 10⁵ = 10 × 10 × 10 × 10 × 10 = 100,000.
• 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 4 × 2 × 2 × 2 × 2 = 8 × 2 × 2 × 2 = 16 × 2 × 2 = 32 × 2 = 64.

This process is fundamental for working with calculations involving powers. Always double-check by meticulously performing each multiplication step. Using a calculator is helpful for larger numbers, but understanding the underlying process is crucial.

What is Index Notation?

Index notation is a concise way to represent repeated multiplication. It consists of a base number and an index (also called an exponent or power). The base is the number being multiplied, and the index indicates how many times the base is multiplied by itself. For example, in 5^3, 5 is the base and 3 is the index. This means 5 x 5 x 5. It's important to differentiate this from 5 x 3, which is a common mistake. Understanding the concept of repeated multiplication is key to correctly interpreting and calculating values with indices.

Calculating Values from Index Notation

To calculate the value of a number written in index notation, you perform the repeated multiplication. For example, to calculate 4^3:

1. Identify the base: 4
2. Identify the index: 3
3. Multiply the base by itself the number of times indicated by the index: 4 x 4 x 4.
4. Perform the multiplication: 4 x 4 = 16. Then 16 x 4 = 64. So, 4^3 = 64.

Always show the steps of multiplication, especially for larger indices or when working with bigger base numbers, to avoid simple calculation errors.

Special Cases: Index of 1 and 0

Any number raised to the power of 1 is just the number itself. For example, 7^1 = 7. This makes sense because it means 7 is multiplied by itself '1 time' (or simply appears once). Any non-zero number raised to the power of 0 is equal to 1. For example, 5^0 = 1 and 100^0 = 1. This rule is a fundamental property of indices and is often used in simplifying expressions. Note that 0^0 is undefined in this context at this level.

Understanding Negative Indices

A negative index indicates the reciprocal of the base raised to the positive power. That is, a⁻ⁿ = 1/aⁿ. For example, 5⁻² = 1/5² = 1/25. This rule is particularly useful when dealing with fractions or when moving terms between the numerator and denominator of an expression. When you encounter a negative index, immediately think 'one over the positive power'.

Mastering Fractional Indices

Fractional indices connect powers and roots. Specifically, a¹/ⁿ = ⁿ√a, meaning the nth root of 'a'. For example, 16¹/² = √16 = 4, and 27¹/³ = ³√27 = 3. When the numerator is not 1, as in aᵐ/ⁿ, it means the nth root of 'a' raised to the power of 'm', or (ⁿ√a)ᵐ. This can also be interpreted as the mth power of 'a' then taking the nth root, (aᵐ)¹/ⁿ. For instance, 8²/³ = (³√8)² = 2² = 4. Remember that the denominator indicates the root and the numerator indicates the power.

Zero Index Rule

Any non-zero number raised to the power of zero is always 1. That is, a⁰ = 1 (where a ≠ 0). For example, 7⁰ = 1, (x²)⁰ = 1, and (3/4)⁰ = 1. This rule applies universally, regardless of the complexity of the base, as long as the base itself is not zero. It's a key rule for simplifying expressions.

Combining Index Rules for Simplification

Higher level problems often require combining multiple index rules. For example, an expression like (x⁻²y³)¹/² / (x⁵y⁻¹) can be simplified by applying the power of a power rule, then product/quotient rules, and finally dealing with negative indices. Always work systematically: simplify powers raised to powers first, then handle multiplication and division of terms with the same base, and finally, resolve any negative indices into their reciprocal forms.

What are Powers?

A power (also called an index or exponent) indicates how many times a base number is multiplied by itself. For example, in 3⁴, 3 is the base and 4 is the exponent. This means 3 × 3 × 3 × 3. The result is 81. Remember that any number raised to the power of 0 is 1 (e.g., 7⁰ = 1) and any number raised to the power of 1 is itself (e.g., 7¹ = 7). Commonly used powers include 'squared' (power of 2, like 5² = 25) and 'cubed' (power of 3, like 2³ = 8).

Understanding Roots

Roots are the inverse operations of powers. The square root of a number (√) is a value that, when multiplied by itself, gives the original number. For example, √49 = 7 because 7 × 7 = 49. Every positive number has two square roots (a positive and a negative one, e.g., ±7), but typically we refer to the positive one. The cube root of a number (³√) is a value that, when multiplied by itself three times, gives the original number. For example, ³√64 = 4 because 4 × 4 × 4 = 64. You can take the cube root of negative numbers, e.g., ³√-8 = -2.

Calculation of Powers

To calculate powers, simply multiply the base number by itself the number of times indicated by the exponent. For instance, to calculate 6³, you do 6 × 6 × 6. First, 6 × 6 = 36. Then, 36 × 6 = 216. So, 6³ = 216. Make sure to be careful with the order of operations if powers are part of a larger expression (BODMAS/PEMDAS). Practice with a variety of small numbers to become fluent.

Estimation of Roots

When you need to estimate a square root or cube root for a number that isn't a perfect square or cube, identify the two perfect squares or cubes it lies between. For example, to estimate √30: you know 5² = 25 and 6² = 36. So, √30 is between 5 and 6, and a bit closer to 5 than 6. Similarly, for ³√100: you know 4³ = 64 and 5³ = 125. So, ³√100 is between 4 and 5, and closer to 5. This method helps to approximate the value without a calculator.