Algebraic Expressions and Index Laws Revision Notes for HSC SSCE Mathematics Standard

Algebraic Expressions and Index Laws

Understanding algebraic expressions

An algebraic expression uses letters (called pronumerals) to represent numbers. These letters can stand for unknown values or values that change. For example, in the equation $x + 5 = 8$ , the pronumeral $x$ represents an unknown number that we can work out (in this case, $x = 3$ ).

When working with algebraic expressions, you'll encounter different types of terms. The number in front of a pronumeral is called a coefficient. Understanding how these parts work together is essential for simplifying expressions and solving problems.

Like terms and collecting them

Like terms are terms that contain exactly the same pronumerals. For example, $2a$ and $5a$ are like terms because they both contain just the letter $a$ . However, $2a$ and $2b$ are not like terms because they contain different letters.

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You can only add or subtract like terms. When you do this, you're essentially combining the coefficients whilst keeping the pronumeral part the same.

This process is called collecting like terms or simplifying the algebraic expression.

Steps for adding and subtracting like terms:

Identify the like terms (terms with exactly the same pronumerals)
Remember that only like terms can be added or subtracted
Add or subtract the coefficients (the numbers before the pronumerals)

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Worked Example: Simplifying Like Terms

Simplify $5x^2 + 6x - 3x^2 - x$

Step 1: Group the like terms together $5x^2 - 3x^2 + 6x - x$

Step 2: Add and subtract coefficients $(5 - 3)x^2 + (6 - 1)x$

Step 3: Calculate $2x^2 + 5x$

Final answer: $2x^2 + 5x$

Working with algebraic fractions

An algebraic fraction is a fraction that contains pronumerals. When adding or subtracting algebraic fractions, you follow similar steps to those used with numerical fractions.

Steps for adding and subtracting algebraic fractions:

Find a common denominator (preferably the lowest common denominator)
Express each fraction as an equivalent fraction with the common denominator
Simplify the numerator by adding or subtracting the like terms

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Worked Example: Adding Algebraic Fractions

Write $\frac{2a}{3} - \frac{a}{6}$ as a single fraction

Step 1: Find the lowest common denominator The lowest common denominator of $3$ and $6$ is 6

Step 2: Convert to equivalent fractions $\frac{2a \times 2}{3 \times 2} - \frac{a}{6} = \frac{4a}{6} - \frac{a}{6}$

Step 3: Subtract the numerators $\frac{4a - a}{6} = \frac{3a}{6}$

Step 4: Simplify $\frac{a}{2}$

Final answer: $\frac{a}{2}$

Index laws

Understanding index notation

Index form or index notation is a shorthand way of writing repeated multiplication. Instead of writing $a \times a$ , we can write $a^2$ . This makes mathematical expressions much easier to read and work with.

When a number or pronumeral is expressed in index form, the index (also called the power or exponent) tells you how many times the multiplication occurs. For example:

$a^n = a \times a \times a \times ... \times a \text{ (n factors)}$

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In the expression $a^n$ :

$a$ is called the base
$n$ is the index or power

The four main index laws

These laws are fundamental rules that help you work with expressions written in index form. They make calculations much simpler and more efficient.

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Index Law 1: Multiplying with the same base

$a^m \times a^n = a^{m+n}$

When multiplying terms with the same base, add the indices together.

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Index Law 2: Dividing with the same base

$a^m \div a^n = a^{m-n}$

or equivalently: $\frac{a^m}{a^n} = a^{m-n}$

When dividing terms with the same base, subtract the indices.

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Index Law 3: Power of a power

$(a^m)^n = a^{mn}$

When raising a term in index form to another power, multiply the indices.

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Index Law 4: The zero index

$a^0 = 1$

Any term raised to the power of zero equals one.

Multiplying algebraic terms using index laws

When multiplying algebraic terms, you need to work systematically through the coefficients and pronumerals separately. This ensures you don't miss any steps and arrive at the correct answer.

Steps for multiplying and dividing algebraic terms:

Write the expression in expanded form
If dealing with fractions, cancel any common factors first
Multiply and divide the coefficients
Multiply and divide the pronumerals using index laws
Write the coefficient before the pronumerals
Write pronumerals in alphabetical order and express in index form

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Worked Example: Multiplying Algebraic Terms

Simplify $5y^2 \times 4y^3$

Step 1: Write in expanded form $5 \times y^2 \times 4 \times y^3$

Step 2: Multiply coefficients $5 \times 4 = 20$

Step 3: Multiply pronumerals using index laws $y^2 \times y^3 = y^{2+3} = y^5$

Final answer: $20y^5$

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Worked Example: Multiple Pronumerals

Simplify $ab^3 \times (-2a^5b)$

Step 1: Write in expanded form $1 \times a \times b^3 \times (-2) \times a^5 \times b$

Step 2: Multiply coefficients $1 \times (-2) = -2$

Step 3: Multiply pronumerals

$a \times a^5 = a^6$
$b^3 \times b = b^4$

Step 4: Write in alphabetical order

Final answer: $-2a^6b^4$

Dividing algebraic terms using index laws

Division follows similar principles to multiplication, but you subtract indices instead of adding them when the bases are the same.

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Worked Example: Dividing Algebraic Terms

Simplify $10d^{12} \div 5d^3$

Step 1: Write in expanded form $10 \div 5 \times d^{12} \div d^3$

Step 2: Divide coefficients $10 \div 5 = 2$

Step 3: Divide pronumerals using index laws $d^{12} \div d^3 = d^{12-3} = d^9$

Final answer: $2d^9$

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Worked Example: Fractions

Simplify $\frac{21m^2n}{7m}$

Step 1: Write in expanded form $\frac{21 \times m \times m \times n}{7 \times m}$

Step 2: Divide coefficients $21 \div 7 = 3$

Step 3: Cancel common factors $m$ appears once in numerator and once in denominator, so: $m^2 \div m = m$

Final answer: $3mn$

Applying index laws with powers

Sometimes you need to apply multiple index laws to simplify a single expression. This is particularly common when dealing with expressions raised to powers.

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Worked Example: Zero Index

Simplify $2x^0 + 5$

Step 1: Apply the zero index law $x^0 = 1$

Step 2: Substitute $2 \times 1 + 5$

Step 3: Evaluate $2 + 5 = 7$

Final answer: $7$

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Worked Example: Power of a Power

Simplify $(6pq^3)^2$

Step 1: Apply the power to each part $(6)^2 \times (p)^2 \times (q^3)^2$

Step 2: Calculate each component

$(6)^2 = 36$
$(p)^2 = p^2$
$(q^3)^2 = q^6$

Step 3: Combine $36 \times p^2 \times q^6$

Final answer: $36p^2q^6$

Simplifying complex algebraic fractions

When working with fractions containing algebraic terms, look for opportunities to cancel common factors before multiplying. This makes the calculation much simpler.

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Worked Example: Complex Algebraic Fractions

Simplify $\frac{x^4}{6y^2} \times \frac{4xy^5}{10}$

Step 1: Write in expanded form $\frac{x^4}{6y^2} \times \frac{4xy^5}{10}$

Step 2: Identify and cancel common factors $2$ is common to $4$ , $6$ , and $10$ : $\frac{x^4}{3y^2} \times \frac{2xy^5}{5}$

Step 3: Cancel $y^2$ from numerator and denominator $\frac{x^4}{3} \times \frac{2xy^3}{5}$

Step 4: Multiply numerators $x^4 \times x = x^5$

Step 5: Multiply denominators $3 \times 5 = 15$

Final answer: $\frac{x^5y^3}{15}$ or $\frac{1}{15}x^5y^3$

Expanding algebraic expressions

Understanding the distributive law

Grouping symbols such as parentheses ( ) and brackets [ ] indicate which operations should be performed first. These symbols can be removed using either the order of operations (working out what's inside first) or the distributive law (multiplying everything inside by what's outside).

The distributive law states that when you have a number or term multiplied by an expression in brackets, you must multiply that number or term by each part inside the brackets.

To expand an algebraic expression means to remove the grouping symbols by applying the distributive law. After expanding, you should simplify the result by collecting any like terms.

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The distributive law:

$a(b + c) = a \times b + a \times c = ab + ac$

$a(b - c) = a \times b - a \times c = ab - ac$

Steps for expanding algebraic expressions:

Multiply the number or term outside the grouping symbol by:
- a) the first term inside the grouping symbol
- b) the second term inside the grouping symbol
Simplify and collect like terms if required

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You must multiply all terms inside the brackets by the term outside. A common mistake is to forget to multiply the second (or subsequent) terms.

Expanding with positive coefficients

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Worked Example: Positive Coefficient

Expand $5(2y - 3)$

Step 1: Write in expanded form $5 \times (2y - 3)$

Step 2: Multiply first term $5 \times 2y = 10y$

Step 3: Multiply second term $5 \times (-3) = -15$

Final answer: $10y - 15$

Expanding with negative coefficients

When the term outside the brackets is negative, you need to be particularly careful with signs. Remember that a negative multiplied by a negative gives a positive.

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Worked Example: Negative Coefficient

Expand $-(m - 5)$

Step 1: Recognize this as $-1 \times (m - 5)$

Step 2: Multiply first term $-1 \times m = -m$

Step 3: Multiply second term $-1 \times (-5) = +5$

Final answer: $-m + 5$

Expanding and simplifying multiple expressions

When you have more than one set of brackets in an expression, expand each set separately and then combine the results by collecting like terms.

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Worked Example: Multiple Brackets

Remove the grouping symbols for $2(3x + 4) + 3(x - 1)$ and simplify

Step 1: Expand first bracket $2 \times 3x + 2 \times 4 = 6x + 8$

Step 2: Expand second bracket $3 \times x + 3 \times (-1) = 3x - 3$

Step 3: Combine $6x + 8 + 3x - 3$

Step 4: Collect like terms $(6x + 3x) + (8 - 3)$

Final answer: $9x + 5$

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Worked Example: Pronumeral Outside Brackets

Expand $y(5y + 1) - y(y - 6)$ and simplify

Step 1: Expand first bracket $y \times 5y + y \times 1 = 5y^2 + y$

Step 2: Expand second bracket $y \times y - y \times 6 = y^2 - 6y$

Step 3: Combine (note the minus sign before the second bracket) $5y^2 + y - y^2 + 6y$

Step 4: Collect like terms $(5y^2 - y^2) + (y + 6y)$

Final answer: $4y^2 + 7y$

Remember!

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Key Points to Remember:

Like terms have exactly the same pronumerals and can be added or subtracted by combining their coefficients
When working with algebraic fractions, find a common denominator before adding or subtracting
The four index laws are essential tools:
- Add indices when multiplying
- Subtract when dividing
- Multiply indices for powers of powers
- Anything to the power of zero equals one
Always write pronumerals in alphabetical order in your final answer
The distributive law requires you to multiply everything inside the brackets by the term outside - don't forget any terms!

Algebraic Expressions and Index Laws (HSC SSCE Mathematics Standard): Revision Notes