Algebraic Roots And Indices

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Notes

Laws of Indices

**a¹ = a**: Any number to the power 1 is itself.
**a⁰ = 1**: Any non‑zero number to the power 0 equals 1.
**aᵐ × aⁿ = aᵐ⁺ⁿ**: To multiply same bases, add the powers.
**aᵐ ÷ aⁿ = aᵐ⁻ⁿ**: To divide same bases, subtract the powers.
**(aᵐ)ⁿ = aᵐⁿ**: To raise a power to another power, multiply the powers.
**(ab)ⁿ = aⁿbⁿ**: To raise a product to a power, apply the power to each factor.
**(a/b)ⁿ = aⁿ/bⁿ**: To raise a fraction to a power, apply the power to numerator and denominator.

Negative and Fractional Indices

**a⁻ⁿ = 1/aⁿ**: A negative power means the reciprocal.
**(a/b)⁻ⁿ = (b/a)ⁿ = bⁿ/aⁿ**: Reciprocal of a fraction to a positive power.
**a^(1/n) = ⁿ√a**: The fractional power 1/n is the nth root.
**a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)**: The power m/n means nth root then mth power (or vice versa).
**a^(-1/n) = 1/ⁿ√a**: Negative fractional power gives one over a root.

Simplifying Expressions with Indices

Work out the number part and the algebra part separately.
Example: (3x⁷) × (6x⁴) = 18x¹¹.
Example: 6x⁷ ÷ 3x⁴ $= 2x^{3}$ .
Example: (3x⁷)² = 9x¹⁴.
Always combine like terms using the index laws.

Solving Equations with Unknown Powers

If both sides have the same base, set the powers equal.
Example: 4³ˣ = 4⁹ ⇒ $3x = 9$ ⇒ $x = 3$ .
Simplify first if needed: 3²ˣ × 3⁴ = 3¹⁸ ⇒ $2x + 4 = 18$ ⇒ $x = 7$ .
Use index laws to rewrite both sides with a common base.

Worked Example: Simplifying Powers

Simplify (u⁵)⁵: use (aᵐ)ⁿ = aᵐⁿ → u²⁵.
Simplify $(q^{2} \times$ q⁵)/q¹⁰: numerator = q⁷, then q⁷⁻¹⁰ = q⁻³.
Hence qˣ = q⁻³ ⇒ $x = -3$ .

Worked Example: Fractional and Negative Powers

Rewrite 1/∛x⁴ as xⁿ: ∛x⁴ $= x^(\frac{4}{3})$ , so $\frac{1}{x}^(\frac{4}{3}) = x^(-\frac{4}{3})$ .
Find m and a in (ax⁶)^(1/m) $= 8x^{3}$ : apply power → $a^(\frac{1}{m}) x^(\frac{6}{m}) = 8x^{3}$ .
Equate x powers: $\frac{6}{m} = 3$ ⇒ $m = 2$ .
Equate constants: $a^(\frac{1}{2}) = 8$ ⇒ $\sqrt{a} = 8$ ⇒ $a = 64$ .

Common Exam Question Types

Simplify products and quotients: $e.g., 2x^{2} \times$ 5x⁵ = 10x⁷.
Simplify powers of powers: $e.g., (3w^{3})^{3} =$ 27w⁹.
Simplify expressions with negative indices: e.g., (4/x)⁻² $= x^{2}/16$ .
Solve equations like 2ᵖ = 1/8⁴: rewrite 8⁴ = 2¹², so 2ᵖ = 2⁻¹² ⇒ $p = -12$ .
Use fractional indices: e.g., (27x⁹)^(2/3) = (∛27x⁹)² $= (3x^{3})^{2} =$ 9x⁶.

Key Tips for Exams

Always simplify step by step, applying one law at a time.
Check if the base can be expressed as a power of a smaller number.
For negative powers, take the reciprocal before applying positive power.
For fractional powers, remember root then power (or power then root).
In equations with the same base, equate the exponents.

Index Laws Summary

Negative and Fractional Indices Examples

Solving Equations with Indices

Simplifying Expressions Worked Example

Practice questions

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1.Simplify $t^{21} \div t^{7}$ .
Easy
- A $t^{14}$
- B $t^{28}$
- C $t^{3}$
- D $t^{147}$
2.Simplify $(u^{5})^{5}$ .
Easy
- A $u^{25}$
- B $u^{10}$
- C $u^{5}$
- D $u^{1}$
3.Simplify $(x^{8})^{3}$ .
Easy
- A $x^{24}$
- B $x^{11}$
- C $x^{8}$
- D $x^{3}$
4. $t^{x} \times t^{2} = t^{10}$ . Find the value of x.
Easy
- A8
- B5
- C12
- D20
5.Simplify fully $(3e)^{0}$ .
Easy
- A1
- B0
- C3
- D3e
6.Simplify $w^{2} \times w^{3}$ .
Easy
- A $w^{5}$
- B $w^{6}$
- C $w^{1}$
- D $w^\frac{2}{3}$
7.Simplify $(3x^{2} y^{4})^{3}$ .
Medium
- A $27 x^{6} y^{12}$
- B $9 x^{5} y^{7}$
- C $27 x^{5} y^{7}$
- D $9 x^{6} y^{12}$
8.Simplify $2x^{2} \times 5x^{5}$ .
Medium
- A $10 x^{7}$
- B $10 x^{10}$
- C $7 x^{7}$
- D $7 x^{10}$

Summary

## Algebraic Roots & Indices ### Laws of Indices - **Multiplication**: $a^m \times a^n = a^{m+n}$ - **Division**: $a^m \div a^n = a^{m-n}$ - **Power of a power**: $(a^m)^n = a^{mn}$ - **Power of a product**: $(ab)^n = a^n b^n$ - **Power of a fraction**: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ - **Zero exponent**: $a^0 = 1$ - **Negative exponent**: $a^{-n} = \frac{1}{a^n}$ - **Fractional exponent**: $a^{\frac{1}{n}} = \sqrt[n]{a}$, $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ - **Negative fractional**: $a^{-\frac{1}{n}} = \frac{1}{\sqrt[n]{a}}$ ### Simplifying Expressions - Work out number and algebra parts separately. - Example: $(3x^7) \times (6x^4) = 18x^{11}$ - Example: $\frac{6x^7}{3x^4} = 2x^3$ - Example: $(3x^7)^2 = 9x^{14}$ ### Solving Equations with Unknown Powers - If both sides have the same base, equate the exponents. - Example: $4^{3x} = 4^9 \Rightarrow 3x = 9 \Rightarrow x = 3$ - Simplify first if needed: $3^{2x} \times 3^4 = 3^{18} \Rightarrow 3^{2x+4} = 3^{18} \Rightarrow 2x+4=18 \Rightarrow x=7$ ### Worked Examples 1. Simplify $(u^5)^5 = u^{25}$ 2. If $q^x = \frac{q^2 \times q^5}{q^{10}}$, find $x$. Simplify numerator: $q^7$, then $\frac{q^7}{q^{10}} = q^{-3}$, so $x = -3$. 3. Rewrite $\frac{1}{\sqrt[3]{x^4}}$ as $x^n$. $\sqrt[3]{x^4} = x^{4/3}$, so $\frac{1}{x^{4/3}} = x^{-4/3}$. 4. Find constants $m$ and $a$ given $(ax^6)^{1/m} = 8x^3$. Expand: $a^{1/m} x^{6/m} = 8x^3$. Equate $x$ powers: $6/m = 3 \Rightarrow m=2$. Then $a^{1/2}=8 \Rightarrow a=64$. ### Key Tips - Always simplify using index laws before solving. - For negative fractional powers, take reciprocal first. - When equating, match coefficients and powers separately.

Key terms

Index Laws: Rules for simplifying expressions with powers: multiplication (add exponents), division (subtract exponents), power of a power (multiply exponents).
Negative Index: A negative exponent indicates the reciprocal: $a^{-n} = \frac{1}{a^n}$.
Fractional Index: A fractional exponent $\frac{m}{n}$ means the nth root raised to the mth power: $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.
Zero Index: Any non-zero number to the power of zero equals 1: $a^0 = 1$.
Equating Exponents: If $a^m = a^n$ and $a>0, a\neq1$, then $m=n$. Used to solve for unknown exponents.

Slides

Laws of Indices

Index laws apply to both numbers and algebra.
Key rules: $a^{m} \times a^{n} = a^(m+n), a^{m} \div a^{n} = a^(m-n), (a^{m})^{n} = a^(mn)$ .
Also: $a^{0} = 1, a^(-n) = \frac{1}{a}^{n}, a^(\frac{1}{n}) = n\sqrt{a}$ .

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

$u^{10}$
$u^{25}$
$u^{0}$
$u^{5}$

Multiplying and Dividing Indices

To multiply indices with the same base, add the powers: $a^{m} \times a^{n} = a^(m+n)$ .
To divide indices with the same base, subtract the powers: $a^{m} \div a^{n} = a^(m-n)$ .
Example: $t^{21} \div t^{7} = t^(21-7) = t^{14}$ .

Simplify: $x^{8} \times x^{3} = x^{____}$

Power of a Product and Fraction

$(ab)^{n} = a^{n} b^{n}$ : apply the power to each factor.
$(\frac{a}{b})^{n} = a^{n} / b^{n}$ : apply the power to numerator and denominator.
Example: $(3x^{2})^{3} = 27x^{6}$ .

Which of these are true?

$(5x^{2})^{3} = 125x^{6}$
$(2y^{3})^{4} = 16y^{12}$
$(\frac{a}{b})^{2} = a^\frac{2}{b}^{2}$
$(3e)^{0} = 3e$

Zero and Negative Powers

Any non-zero number to the power 0 equals $1: a^{0} = 1$ .
A negative power means reciprocal: $a^(-n) = \frac{1}{a}^{n}$ .
Example: $(3e)^{0} = 1$ .

Simplify $(3e)^{0}$

Fractional Powers

$a^(\frac{1}{n}) = n\sqrt{a} (the$ nth root).
$a^(\frac{m}{n}) = (n\sqrt{a})^{m} = n\sqrt{a^{m}}$ .
Example: $27^(\frac{1}{3}) = 3\sqrt{27} = 3$ .

Simplify: $(27x^{6})^(\frac{1}{3}) = ____ x^{2}$

Negative Fractional Powers

$a^(-\frac{m}{n}) = 1 / a^(\frac{m}{n})$ .
Alternatively, $a^(-\frac{m}{n}) = 1 / (n\sqrt{a})^{m}$ .
Example: $8^(-\frac{2}{3}) = 1 / (8^(\frac{2}{3})) = \frac{1}{4}$ .

Simplify $(64)^(-\frac{1}{2})$

8
$\frac{1}{8}$
64
$\frac{1}{64}$

Solving Equations with Indices

If both sides have the same base, set the exponents equal.
Simplify first using index laws if needed.
Example: $2^(3x) = 2^{9}$ ⇒ $3x=9$ ⇒ $x=3$ .

If $t^{x} \times t^{2} = t^{10}$ , then $x = ____$

Combining Laws

Simplify expressions by handling numbers and algebra separately.
Example: $2x^{3} \times 3x^{2} = 6x^{5}$ .
Example: $(5x^{4})^{3} = 125x^{12}$ .

Which simplifications are correct?

$2x^{2} \times 5x^{5} = 10x^{7}$
$(4xy^{2})^{3} = 64x^{3}y^{6}$
$a^{3} \times a^{6} = a^{18}$
$(3w^{3})^{3} = 27w^{9}$

Harder Fractional Powers

For $a^(\frac{m}{n})$ , take the nth root first or raise to m first.
Example: $(27x^{9})^(\frac{2}{3}) = (27^(\frac{1}{3}))^{2} \times (x^{9})^(\frac{2}{3}) = 3^{2} \times x^{6} = 9x^{6}$ .
Be careful with negative fractional powers: take reciprocal first.

Simplify $(27x^{9})^(\frac{2}{3})$

$9x^{6}$
$3x^{6}$
$9x^{9}$
$27x^{6}$

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