Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

Simplifying Expressions and Solving Index Equations

Updated 10 October 2026 · Fact-checked

Simplifying with indices means using the laws of indices to combine powers into one form. To solve an index equation, write both sides with the same base, then equate the powers and solve for x. Always check the answer by substituting it back into the equation.

Understand Simplifying Expressions and Solving Index Equations

An index (or exponent) tells you how many times a base is multiplied by itself. In 2⁵, the base is 2 and the index is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Simplifying an expression means using the laws of indices to reduce it to a neater form. You multiply powers of the same base by adding indices, divide by subtracting, and raise a power to a power by multiplying. A surd is a root written with a radical sign, and it is just a fractional index: √a = a^(1/2) and ∛a = a^(1/3). Converting surds to fractional indices lets you use the same laws.

An index equation has the unknown in the exponent, like 3^(x+1) = 81. The key idea is simple. If two powers of the same base (base positive and not equal to 1) are equal, their indices must be equal. So you rewrite both sides with one common base, then drop the base and solve the plain equation in x.

Sometimes the indices match but the bases are different, such as x⁵ = 2⁵. Then you equate the bases. The rule is: if xⁿ = yⁿ with n ≠ 0, then x = y when x and y are positive, or when n is odd. If n is even and the signs are not restricted, then x = ±y. For example, x² = 2² gives x = 2 or x = −2. Most exam questions are built so that one of these two moves solves them.

Key formulas to remember

Product law: aᵐ × aⁿ = a^(m+n); Same base only. Add the indices.
Quotient law: aᵐ ÷ aⁿ = a^(m−n); Same base, a ≠ 0. Subtract the indices.
Power of a power: (aᵐ)ⁿ = a^(mn); Multiply the indices.
Power of a product: (ab)ⁿ = aⁿ bⁿ; Also (a/b)ⁿ = aⁿ ÷ bⁿ for b ≠ 0.
Zero and negative index: a⁰ = 1 and a^(−n) = 1 ÷ aⁿ; For a ≠ 0.
Fractional index: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ; Gives the link between surds and indices.
Equating powers: If aˣ = aʸ (a > 0, a ≠ 1), then x = y; The main tool for index equations.
Equating bases: If xⁿ = yⁿ with x, y > 0 and n ≠ 0, then x = y; Use when the indices are already equal. If n is odd, positivity is not needed. If n is even and signs are unrestricted, x = ±y.

How to solve Simplifying Expressions and Solving Index Equations questions

Use this method for any simplification or index equation question.

1Convert all roots (surds) to fractional indices and all negative indices to reciprocals.
2Break each number into prime factors, such as 8 = 2³, 27 = 3³, 125 = 5³.
3Apply the laws of indices to combine terms with the same base into a single power on each side.
4For an equation, make both sides use the same base.
5Equate the indices and form a simple equation in x.
6Solve the equation for x. If the unknown is in a power of a power, multiply indices first.
7Substitute your value back to check both sides match.
8Match the result with the options.

Quickest way: Prime-base and option-check method

When to use it: Use it in the 1-hour objective paper when the numbers are small, such as powers of 2, 3 and 5.

Write the bases as small primes straight away: 4 = 2², 9 = 3², 16 = 2⁴, 81 = 3⁴.
Equate indices mentally and solve the linear equation.
If the equation looks hard, substitute each option into it. Test the easiest option first.
For expressions, cancel common factors before expanding anything.
Stop as soon as one option satisfies the equation, since only one is correct.

Common mistakes in Simplifying Expressions and Solving Index Equations

Adding indices when the bases are different, such as 2³ × 3² = 6⁵.
Students remember 'add the powers' but forget the bases must be the same.
Fix: Add indices only for the same base. Here 2³ × 3² = 8 × 9 = 72.
Writing (aᵐ)ⁿ = a^(m+n).
It gets mixed up with the product law.
Fix: A power raised to a power multiplies: (2³)² = 2⁶ = 64, not 2⁵.
Treating a^(−n) as a negative number.
The minus sign is read as the sign of the value.
Fix: A negative index means reciprocal. 2⁻³ = 1/8, which is positive.
Equating only part of the index, e.g. using 2x = 5 instead of 2x + 1 = 5.
Students equate only part of the index and ignore the rest of it.
Fix: Equate the entire index: if 3^(2x+1) = 3⁵, then 2x + 1 = 5, so x = 2.
Reducing a^(m/n) wrongly by flipping m and n.
Students mix up which number is the power and which is the root.
Fix: The denominator is the root and the numerator is the power. 8^(2/3) = (∛8)² = 4.
Not breaking numbers into primes, so bases never match.
Students try to equate 4^x with 2^5 directly.
Fix: Rewrite 4 as 2² first. Then 2^(2x) = 2⁵ gives x = 5/2.

Worked examples

Example 1

Find x if 4^(x+1) = 8^(x−1).

Show the solution

Write both bases as powers of 2: 4 = 2² and 8 = 2³.
Left side: 4^(x+1) = (2²)^(x+1) = 2^(2x+2).
Right side: 8^(x−1) = (2³)^(x−1) = 2^(3x−3).
Equate the indices: 2x + 2 = 3x − 3.
Solve: 2 + 3 = 3x − 2x, so x = 5.
Check: 4⁶ = 4096 and 8⁴ = 4096. Both sides match.

Answer: x = 5

Example 2

Simplify (8^(2/3) × 9^(1/2)) ÷ 27^(1/3).

Show the solution

8^(2/3) = (∛8)² = 2² = 4.
9^(1/2) = √9 = 3.
27^(1/3) = ∛27 = 3.
Numerator = 4 × 3 = 12.
Divide: 12 ÷ 3 = 4.

Answer: 4

Exam tips

Memorise squares and cubes up to 15 and powers of 2, 3 and 5. They make prime-base conversion instant.
Questions often hide the same base, such as 4, 8, 16 for base 2 or 9, 27, 81 for base 3. Spot the family first.
If the algebra gets messy, test the four options. With no negative marking, a quick substitution is safe and fast.
Watch for equations like 2^(x+2) − 2^x = 24. Take 2^x common: 2^x(4 − 1) = 24, so 2^x = 8 and x = 3.
Do not leave any question blank. Every question is 2 marks and there is no negative marking.

Practice questions from Indices and Logarithms

Simplifying Expressions and Solving Index Equations: frequently asked questions

How do I solve an exponential equation with the same base?

Once both sides have the same base (positive and not 1), drop the base and equate the indices. Then solve the resulting equation for x. For example, 5^(x−2) = 5⁴ gives x − 2 = 4, so x = 6.

What if the bases are different in an index equation?

Try to rewrite both bases as powers of one common number, usually a prime. For example, 4 and 8 both become powers of 2. If the indices are already equal, you can equate the bases instead.

How do surds connect to indices?

A surd is a root, and a root is a fractional index. So √a = a^(1/2) and ∛a = a^(1/3). Converting surds to fractional indices lets you use the normal laws of indices.

Can I solve index questions by checking the options?

Yes. The paper is objective with four options and no negative marking, so substituting options is a valid shortcut. Start with the simplest value, and stop when one option makes both sides equal.