Equations Involving Indices Revision Notes for Leaving Cert Mathematics

Equations Involving Indices

Understanding the basic principle

Equations involving indices are mathematical equations where the unknown variable appears as an exponent (power). These equations can be solved using a fundamental rule that makes the process straightforward once you understand the method.

The key principle for solving equations involving indices is simple but powerful:

When two powers have the same base, their indices must be equal for the equation to be true.

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Fundamental Rule for Index Equations

If $a^x = a^y$ , then $x = y$

This is the foundation for solving all equations involving indices.

For example:

If $5^x = 5^2$ , then $x = 2$
If $7^x = 7^1$ , then $x = 1$

The solution method

To solve equations involving indices, follow these essential steps:

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Step-by-Step Method

Express both sides as powers of the same base number
Apply the fundamental rule to equate the indices
Solve the resulting simple equation

This systematic approach works for all index equations, regardless of complexity.

Let's see how this works with a typical example:

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Worked Example: Solving $25^x = 125$

Step 1: Express both sides using the same base

Notice that both 25 and 125 can be written as powers of 5
$25 = 5^2$ and $125 = 5^3$
So the equation becomes: $(5^2)^x = 5^3$

Step 2: Simplify the left side using the power rule

$(5^2)^x = 5^{2x}$
The equation is now: $5^{2x} = 5^3$

Step 3: Apply the fundamental rule

Since the bases are equal: $2x = 3$
Therefore: $x = \frac{3}{2} = 1\frac{1}{2}$

Working with different types of indices

Positive integer indices

These are the most straightforward cases where you can often recognise the base immediately.

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Example: Solve $4^x = 16$

Express as powers of 2: $(2^2)^x = 2^4$
Simplify: $2^{2x} = 2^4$
Equate indices: $2x = 4$
Solution: $x = 2$

Negative indices

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Key Concept: Negative Indices

Remember that negative indices represent reciprocals: $a^{-n} = \frac{1}{a^n}$

This relationship is crucial for solving equations with fractions.

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Example: Solve $3^x = \frac{1}{27}$

Express the right side with positive indices: $3^x = \frac{1}{3^3} = 3^{-3}$
Apply the fundamental rule: $x = -3$

Fractional indices

When dealing with fractions in the base or result, be careful with your power manipulations.

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Example: Solve $25^x = \frac{1}{125}$

Express as powers of 5: $(5^2)^x = \frac{1}{5^3} = 5^{-3}$
Simplify: $5^{2x} = 5^{-3}$
Equate indices: $2x = -3$
Solution: $x = -1\frac{1}{2}$

Advanced examples with surds

Some equations involve square roots (surds) which need to be expressed as fractional indices.

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Worked Example: Express $\frac{81}{\sqrt{3}}$ as a power of 3, then solve $3^{x-2} = \left(\frac{81}{\sqrt{3}}\right)^2$

Step 1: Express as powers of 3

$81 = 3^4$
$\sqrt{3} = 3^{\frac{1}{2}}$
So $\frac{81}{\sqrt{3}} = \frac{3^4}{3^{\frac{1}{2}}} = 3^{4-\frac{1}{2}} = 3^{\frac{7}{2}}$

Step 2: Substitute into the equation

$3^{x-2} = \left(3^{\frac{7}{2}}\right)^2$
$3^{x-2} = 3^{\frac{7}{2} \times 2} = 3^7$

Step 3: Solve

$x - 2 = 7$
$x = 9$

Key formulas to remember

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Essential Formulas for Index Equations

Fundamental rule: If $a^x = a^y$ , then $x = y$
Negative indices: $a^{-n} = \frac{1}{a^n}$
Power of a power: $(a^m)^n = a^{mn}$
Square roots: $\sqrt{a} = a^{\frac{1}{2}}$
Useful surd relationships: $\sqrt{8} = 2\sqrt{2}$ and $\sqrt{27} = 3\sqrt{3}$

Exam tips and common mistakes

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Critical Points to Avoid Common Errors

Always check if both sides can be expressed with the same base - this is the key to solving these equations
Be careful with negative signs when dealing with negative indices
Remember that $\frac{1}{a^n} = a^{-n}$ - this conversion is essential for many problems
Double-check your arithmetic when working with fractional indices
Look out for surds that can be simplified before expressing as powers

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Key Points to Remember:

The fundamental principle: When bases are equal, indices must be equal
Express both sides as powers of the same base before attempting to solve
Negative indices mean reciprocals: $a^{-n} = \frac{1}{a^n}$
Check your work by substituting your answer back into the original equation
Practice identifying common bases like powers of 2, 3, 5, etc. to solve problems more efficiently

Equations Involving Indices (Leaving Cert Mathematics): Revision Notes