You are halfway through a problem when you hit 2^x \cdot 4^x and your pen freezes. "Product rule says add exponents — but the bases are different. Stuck." That pause is the exact moment where a one-line rewrite would have unlocked the problem. 4 is not really a different base — it is 2^2 in disguise, and once you rewrite it that way, the product rule applies and the expression simplifies in two short steps.

Before you apply any exponent rule, run a quick checkpoint: do the bases actually match? If they look different but can be rewritten as powers of the same underlying number, rewrite first. Skipping this step costs you either a wrong answer or a missed simplification. The rewrite is free. The wrong rule is expensive.

The test

When you see a product, quotient, or equation involving powers with different-looking bases, ask one question: can I write both bases as powers of the same prime, or the same expression?

If yes, rewrite them and proceed with a standard rule. If no, maybe the "same exponent" rule applies, or maybe there is no simplification at all. But you cannot know until you have checked. The test takes two seconds.

The 2^x · 4^x example

Look at 2^x \cdot 4^x. Student instinct: "different bases — the product rule needs the same base — so I'm stuck." Pen down.

The fix is to recognise that 4 is not a different base. 4 = 2^2, so 4^x = (2^2)^x = 2^{2x}. Substitute back:

2^x \cdot 4^x \;=\; 2^x \cdot 2^{2x} \;=\; 2^{x + 2x} \;=\; 2^{3x}

You can leave this as 2^{3x}, or notice 2^{3x} = (2^3)^x = 8^x. The simplified answer is 8^x — but only after the rewrite. The rewrite step is what makes the reasoning rigorous. Without it, you are guessing.

Bases you should recognise as same-prime relatives

The whole tactic depends on spotting that two numbers share a prime-power structure. Here is the shortlist to keep in your head:

When two bases in a problem both sit on the same line of this list — like 4 and 8, or 9 and 27, or 25 and 125 — the common-base rewrite is available. When they sit on different lines — like 4 and 9, or 3 and 5 — it is not, and you need a different tactic. The Break the Base Into a Prime Power sibling drills this recognition in more depth.

Worked example 2 — 9^x \cdot 27^y

Both bases are powers of 3: 9 = 3^2 and 27 = 3^3. Rewrite each power:

9^x \cdot 27^y \;=\; (3^2)^x \cdot (3^3)^y \;=\; 3^{2x} \cdot 3^{3y} \;=\; 3^{2x + 3y}

The exponents on the two factors were different (x and y), so the "same exponent, multiply bases" rule does not apply. The product rule does — but only after the rewrite to a common base of 3.

Worked example 3 — solve 4^x = 8^{x-1}

This is an exponential equation. Both sides are powers of 2:

4^x = 8^{x-1} \;\Longrightarrow\; (2^2)^x = (2^3)^{x-1} \;\Longrightarrow\; 2^{2x} = 2^{3(x-1)}

Once the bases are equal, equate the exponents:

2x = 3(x-1) \;\Longrightarrow\; 2x = 3x - 3 \;\Longrightarrow\; x = 3

Check: 4^3 = 64 and 8^{3-1} = 8^2 = 64. Match. Without the common-base rewrite, this equation is almost unsolvable — you cannot isolate x when it sits inside two different-looking exponentials. The rewrite converts a hard exponential equation into a one-line linear equation.

Worked example 4 — when the bases don't share a prime

Consider 3^x \cdot 5^x. Can 3 and 5 be written as powers of the same prime? No — they are both prime, and different. So the rewrite is not available.

Do not force it. Instead, notice the other feature: the exponents match. That fires a different rule — the same-exponent regrouping:

3^x \cdot 5^x \;=\; (3 \cdot 5)^x \;=\; 15^x.

The checkpoint "can I match the bases?" comes first; if no, move on to the next tool.

The decision tree

When you face a product or quotient of powers, run this decision tree in order:

  1. Same base already? Apply the product or quotient rule directly. 2^x \cdot 2^y = 2^{x+y}.
  2. Different bases, but one is a power of the other (or both are powers of a common third number)? Rewrite to the common base, then apply the product or quotient rule. 2^x \cdot 4^x \to 2^x \cdot 2^{2x} = 2^{3x}.
  3. Different bases, but the same exponent? Regroup as a single power of the product of bases. 3^x \cdot 5^x = 15^x.
  4. None of the above? No simplification is possible. Leave the expression as it is.

The four cases are mutually exclusive once you resolve them in order. If case 1 applies, you never reach case 2. If case 2 applies, you never reach case 3. Learn the order, and the decision is mechanical.

What about negative and fractional exponents

The common-base rewrite works for every exponent type — the power-of-a-power rule (b^k)^n = b^{kn} holds for any n.

Example: simplify 16^{1/2} \cdot 8^{1/3}. Both are powers of 2:

16^{1/2} \cdot 8^{1/3} \;=\; (2^4)^{1/2} \cdot (2^3)^{1/3} \;=\; 2^2 \cdot 2^1 \;=\; 2^3 \;=\; 8.

The rewrite unifies the calculation under a single base of 2 and reduces the whole expression to a single integer.

Common student error — "just add the numbers"

A seductive wrong move is to write 2^x \cdot 4^x = 6^x by "adding the bases." This is wrong. Adding bases is not any exponent rule — the product law adds exponents, not bases, and the same-exponent regrouping multiplies bases, not adds. Before writing anything, ask whether the bases are in a shape where a known rule applies. At no point in any valid rule do you add bases.

Recognition drill

For each expression below, state (a) whether the bases can be rewritten to a common base, and (b) the final simplified form.

Closing

Before you apply any exponent rule, take two seconds to ask: can I get the bases to match? If yes, rewrite first — 4 becomes 2^2, 9 becomes 3^2, 27 becomes 3^3 — and then the standard product, quotient, or equate-exponents rule does its job. If no, move to a different rule or accept that no simplification is possible. Rewriting bases is free. Using the wrong rule is expensive.