Indices II.
Extend index rules to algebraic expressions and solve equations using a common base.
Use all types of indices · simplify algebraic powers · solve common-base equations
Keep the base; operate on the indices
The index rules you used for numbers apply to letters because the repeated-factor idea is unchanged. Three factors of x multiplied by two more factors of x give five factors of x.
Keep coefficients and letter powers separate at first. In 2x³ × 4x², the coefficients multiply to 8 while the five factors of x give x⁵. Restrictions still matter: a negative power of x requires x not to be zero.
The laws from Number still apply when indices contain letters. For fractional indices in these examples, take the base to be positive; division and negative powers require a non-zero base.
| Multiply | xaxb = xa+b |
|---|---|
| Divide | xaxb = xa−b |
| Power of a power | (xa)b = xab |
| Zero and negative | x⁰ = 1; x−a = 1xa |
| Fractional | x12 = √x; x23 = (∛x)² |
Simplify 3x² × 4x⁻12.
Multiply coefficients separately from the powers.
Simplify 5x1/2x⁻².
Pause and explainWhy does x³ × x² give x⁵ rather than x⁶?
There are 3 + 2 factors of x altogether. Multiplying indices is the rule for a power raised to another power, not this product.
Use a common base to solve equations
An equation with powers can sometimes be solved by expressing both sides with the same base. Once the base is a positive number other than 1, matching powers requires matching exponents.
For example, 8 can be rewritten as 2³, so 2ˣ = 8 becomes 2ˣ = 2³ and x = 3. First rewrite both sides, then simplify their exponents using index rules. Do not equate exponents while the bases are still different.
Solve 32x = 27.
Write 27 as 3³. Equal powers of the same positive base (other than 1) have equal indices.
Solve 5x+1 = 25x.
x + 1 = 2x, so x = 1
Solve 8x = 4.
Pause and explainWhat first step helps solve 3ˣ = 27?
Write 27 as 3³. Both sides then have base 3, making x = 3 clear.
Your turn. State your base.
Expand a short repeated product if you are unsure which index rule applies. Check whether the operation is multiplication, division or raising a power.
For exponential equations, show the common base explicitly before comparing the exponents. Substitute your final value into the original equation to confirm that both numerical sides agree.
Simplify (2x³)² × x⁻⁴.
Reveal worked answer
Simplify (x53)32 for x > 0.
Reveal worked answer
Solve 9x−1 = 27.