Algebraic Roots And Indices

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Notes

Laws of Indices

**Index laws** apply to both numbers and algebra.
**a¹ = a**: any term to the power 1 is itself.
**a⁰ = 1**: any non‑zero term to the power 0 equals 1.
**aᵐ × aⁿ = aᵐ⁺ⁿ**: add powers when multiplying same base.
**aᵐ ÷ aⁿ = aᵐ⁻ⁿ**: subtract powers when dividing same base.
**(aᵐ)ⁿ = aᵐⁿ**: multiply powers when raising a power to another power.
**(ab)ⁿ = aⁿbⁿ**: apply the power to each factor in a product.
**(a/b)ⁿ = aⁿ / bⁿ**: apply the power to numerator and denominator.

Negative and Zero Indices

**a⁻¹ = 1/a**: a negative index gives the reciprocal.
**a⁻ⁿ = 1/aⁿ**: general rule for negative powers.
**a⁰ = 1** for any $a \ne 0$ .
Example: 2⁻³ $= \frac{1}{2}^{3} = \frac{1}{8}$ .

Simplifying Expressions with Indices

Work out the **number part** and **algebra part** separately.
Multiply coefficients and use index laws for variables.
Example: (3x⁷) × (6x⁴) = 18x¹¹.
Example: 6x⁷ ÷ 3x⁴ $= 2x^{3}$ .
Example: (3x⁷)² = 9x¹⁴.

Solving Equations with Unknown Indices

Write both sides with the **same base**.
Then equate the indices to find the unknown.
Example: 5ˣ $= 125$ → 5ˣ $= 5^{3}$ → $x = 3$ .
Example: 2ˣ $= \frac{1}{8}$ → 2ˣ = 2⁻³ → $x =$ –3.

Worked Examples from Syllabus

Simplify (u⁵)⁵ = u²⁵ using (aᵐ)ⁿ = aᵐⁿ.
If qˣ $= (q^{2} \times$ q⁵)/q¹⁰, simplify numerator: q²⁺⁵ = q⁷.
Then q⁷/q¹⁰ = q⁷⁻¹⁰ = q⁻³, so $x =$ –3.

Common Exam Question Types

Find the value of x in 6¹⁰ × 6ˣ $= 6^{2}$ → $x =$ –8.
Find x in 5ˣ $\times 5^{3} =$ 5¹² → $x = 9$ .
Simplify $p^{2} \times$ p⁴ = p⁶.
Simplify m¹⁵ ÷ m⁵ = m¹⁰.
Simplify (k³)⁵ = k¹⁵.

Medium and Hard Questions

Simplify $5 \times$ x⁰ $= 5 \times 1 = 5$ .
Simplify $2x^{3} \times 3x^{2} =$ 6x⁵.
Simplify 4p⁵q³ × p²q⁻⁴ = 4p⁷q⁻¹.
Simplify 8t⁸ ÷ 4t⁴ = 2t⁴.
Solve 4ʷ $= \frac{1}{16}$ → 4ʷ = 4⁻² → $w =$ –2.

Index Laws Summary

Multiplying and Dividing Indices

Solving Index Equations

Negative Indices

Practice questions

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1.What is the value of any non-zero number raised to the power of 0?
Easy
- A0
- B1
- Cthe number itself
- Dundefined
2.Simplify $p^{2} \times p^{4}$ .
Easy
- A $p^{6}$
- B $p^{8}$
- C $2p^{6}$
- D $p^{2}$
3.Simplify $m^{15} \div m^{5}$ .
Easy
- A $m^{10}$
- B $m^{20}$
- C $m^{3}$
- D $m^{75}$
4.Simplify $(k^{3})^{5}$ .
Easy
- A $k^{8}$
- B $k^{15}$
- C $k^{2}$
- D $k^{125}$
5.If $9^{12} \div 9^{w} = 9^{4}$ , find the value of w.
Easy
- A8
- B16
- C3
- D48
6.Simplify $(w^{5})^{4}$ .
Easy
- A $w^{9}$
- B $w^{20}$
- C $w^{625}$
- D $w^{1}$
7.Simplify $t^{21} \div t^{7}$ .
Easy
- A $t^{14}$
- B $t^{28}$
- C $t^{3}$
- D $t^{147}$
8.Simplify $(u^{5})^{5}$ .
Easy
- A $u^{10}$
- B $u^{25}$
- C $u^{3125}$
- D $u^{0}$

Summary

## Algebraic Roots & Indices ### Laws of Indices (same base) - **Multiplication**: $a^m \times a^n = a^{m+n}$ Example: $p^2 \times p^4 = p^6$ - **Division**: $a^m \div a^n = a^{m-n}$ Example: $m^{15} \div m^5 = m^{10}$ - **Power of a power**: $(a^m)^n = a^{mn}$ Example: $(k^3)^5 = k^{15}$ - **Power of a product**: $(ab)^n = a^n b^n$ - **Power of a fraction**: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ ### Special Indices - $a^1 = a$ - $a^0 = 1$ - $a^{-1} = \frac{1}{a}$, $a^{-n} = \frac{1}{a^n}$ (negative power = reciprocal) ### Simplifying Expressions - Separate number and algebra parts: $3x^7 \times 6x^4 = (3\times6)(x^{7+4}) = 18x^{11}$ $\frac{6x^7}{3x^4} = \frac{6}{3}x^{7-4} = 2x^3$ $(3x^7)^2 = 3^2 x^{14} = 9x^{14}$ ### Solving Equations with Unknown Index - Write both sides with the same base, then equate indices. Example: $5^x = 125 \Rightarrow 5^x = 5^3 \Rightarrow x=3$ Example: $2^x = \frac{1}{8} \Rightarrow 2^x = 2^{-3} \Rightarrow x=-3$ ### Worked Example Simplify $(u^5)^5$: $(u^5)^5 = u^{5\times5} = u^{25}$ If $q^x = \frac{q^2 \times q^5}{q^{10}}$, find $x$: Numerator: $q^2 \times q^5 = q^7$ Division: $\frac{q^7}{q^{10}} = q^{-3}$ So $q^x = q^{-3} \Rightarrow x = -3$

Key terms

Base: The number or variable that is raised to a power (e.g., in $a^n$, $a$ is the base).
Index (plural: indices): The power or exponent to which a base is raised (e.g., in $a^n$, $n$ is the index).
Law of indices: Rules for simplifying expressions involving powers: multiplication (add indices), division (subtract indices), power of a power (multiply indices).
Reciprocal: The multiplicative inverse; $a^{-1} = \frac{1}{a}$.
Simplify: To combine terms using index laws to write an expression in a more compact form.

Slides

Laws of Indices

Index laws are rules for operations with powers.
They work with both numbers and algebra.
Key laws: $a^{1} = a, a^{0} = 1, a^{m} \times a^{n} = a^(m+n), a^{m} \div a^{n} = a^(m-n), (a^{m})^{n} = a^(mn)$ .

What is the value of $5^{0}$ ?

0
1
5
undefined

Multiplying Indices

To multiply indices with the same base, add the powers: $a^{m} \times a^{n} = a^(m+n)$ .
Example: $p^{2} \times p^{4} = p^(2+4) = p^{6}$ .

Simplify: $x^{3} \times x^{6} = x^____$

Dividing Indices

To divide indices with the same base, subtract the powers: $a^{m} \div a^{n} = a^(m-n)$ .
Example: $m^{15} \div m^{5} = m^(15-5) = m^{10}$ .

Simplify $t^{21} \div t^{7}$ .

$t^{28}$
$t^{14}$
$t^{3}$
$t^{147}$

Power of a Power

To raise a power to another power, multiply the powers: $(a^{m})^{n} = a^(mn)$ .
Example: $(k^{3})^{5} = k^(3\times 5) = k^{15}$ .

Simplify: $(u^{5})^{5} = u^____$

Negative Indices

A negative power means the reciprocal: $a^(-n) = 1/(a^{n})$ .
Example: $2^(-3) = 1/(2^{3}) = \frac{1}{8}$ .

Which of these are true?

$5^(-1) = \frac{1}{5}$
$a^(-2) = 1/(a^{2})$
$3^(-2) = -9$

Solving Equations with Unknown Indices

Write both sides with the same base, then equate the powers.
Example: $5^{x} = 125$ → $5^{x} = 5^{3}$ → $x = 3$ .
Example: $2^{x} = \frac{1}{8}$ → $2^{x} = 2^(-3)$ → $x = -3$ .

Find x if $3^{x} = 81$ .

Combining Laws

Simplify expressions by handling numbers and algebra separately.
Example: $3x^{7} \times 6x^{4} = (3\times 6) \times (x^{7} \times x^{4}) = 18x^{11}$ .
Example: $6x^{7} \div 3x^{4} = (6\div 3) \times (x^{7} \div x^{4}) = 2x^{3}$ .

Simplify: $2x^{3} \times 3x^{2} = ____ x^____$

Worked Example

If $q^{x} = (q^{2} \times q^{5}) / q^{10}$ , find x.
Simplify numerator: $q^{2} \times q^{5} = q^{7}$ .
Divide: $q^{7} / q^{10} = q^(7-10) = q^(-3)$ .
So $x = -3$ .

Find x if $2^{x} = \frac{1}{16}$ .

-4
4
-2
2

Zero and One

$a^{1} = a$ for any base a.
$a^{0} = 1$ for any non-zero base a.
Example: $5^{1} = 5, 5^{0} = 1$ .

Which of these are true?

$7^{1} = 7$
$0^{0} = 1$
$x^{0} = 1 (x \ne 0)$

Flashcards

What is the value of any number (or variable) raised to the power of 0?

Any number (or variable) raised to the power of 0 equals 1. For example, $b^{0} = 1$ .

What is the value of any number (or variable) raised to the power of 1?

Any number (or variable) raised to the power of 1 is itself. For example, $x^{1} = x$ .

Law of indices: $a^{m} \times a^{n} =$ ?

$a^{m} \times a^{n} = a^(m+n)$ . To multiply indices with the same base, add their powers.

Law of indices: $a^{m} \div a^{n} =$ ?

$a^{m} \div a^{n} = a^(m-n)$ . To divide indices with the same base, subtract their powers.

Law of indices: $(a^{m})^{n} =$ ?

$(a^{m})^{n} = a^(m\times n)$ . To raise an index to a new power, multiply the powers.

Law of indices: $(ab)^{n} =$ ?

$(ab)^{n} = a^{n} \times b^{n}$ . To raise a product to a power, apply the power to each factor.

Law of indices: $(\frac{a}{b})^{n} =$ ?

$(\frac{a}{b})^{n} = a^{n} / b^{n}$ . To raise a fraction to a power, apply the power to numerator and denominator.

What does a negative index represent?

A negative index represents the reciprocal. $a^(-n) = 1/(a^{n})$ . For example, $k^(-3) = \frac{1}{k}^{3}$ .

Simplify: $p^{2} \times p^{4}$

$p^{2} \times p^{4} = p^(2+4) = p^{6}$ .

Simplify: $m^{15} \div m^{5}$

$m^{15} \div m^{5} = m^(15-5) = m^{10}$ .

Simplify: $(k^{3})^{5}$

$(k^{3})^{5} = k^(3\times 5) = k^{15}$ .

Simplify: $(w^{5})^{4}$

$(w^{5})^{4} = w^(5\times 4) = w^{20}$ .

Simplify: $t^{21} \div t^{7}$

$t^{21} \div t^{7} = t^(21-7) = t^{14}$ .

Simplify: $(u^{5})^{5}$

$(u^{5})^{5} = u^(5\times 5) = u^{25}$ .

Simplify: $(x^{8})^{3}$

$(x^{8})^{3} = x^(8\times 3) = x^{24}$ .

Simplify: $(y^{5})^{3}$

$(y^{5})^{3} = y^(5\times 3) = y^{15}$ .

Simplify: $(x^{5})^{2}$

$(x^{5})^{2} = x^(5\times 2) = x^{10}$ .

If $x^{3} \times x^{6} = x^{m}$ , find m.

$x^{3} \times x^{6} = x^(3+6) = x^{9}$ , so $m = 9$ .

If $(x^{2})^{n} = x^{12}$ , find n.

$(x^{2})^{n} = x^(2n) = x^{12}$ , so $2n = 12$ , hence $n = 6$ .

Solve: $5^{x} = 125$

$125 = 5^{3}$ , so $5^{x} = 5^{3}$ , therefore $x = 3$ .

Solve: $2^{x} = \frac{1}{8}$

$\frac{1}{8} = \frac{1}{2}^{3} = 2^(-3)$ , so $2^{x} = 2^(-3)$ , therefore $x = -3$ .

Simplify: $5 \times x^{0}$

$x^{0} = 1$ , so $5 \times 1 = 5$ .

Simplify: $2x^{3} \times 3x^{2}$

Multiply coefficients: $2\times 3=6$ . Add powers: $x^(3+2)=x^{5}$ . Result: $6x^{5}$ .

Simplify: $5m^{2} \times 2m^{3}$

$5\times 2=10, m^(2+3)=m^{5}$ , so $10m^{5}$ .

Simplify: $4p^{5} q^{3} \times p^{2} q^(-4)$

$4 \times 1 =$ 4; p^(5+2)=p⁷; $q^(3+(-4))=q^(-1)=\frac{1}{q}$ . Result: $4p^\frac{7}{q}$ .

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