Three short rules do all the heavy lifting: when bases are multiplied you add exponents; when a power is raised to a power you multiply them; when a product is raised to a power, each factor takes the exponent.
In Lesson 7.2 you met the power: a5 is just shorthand for a multiplied by itself five times. That little raised number — the exponent — is a counter. It does not tell you to multiply by 5; it tells you how many copies of the base are being multiplied together. Hold onto that one idea, because every rule in this lesson falls out of it the moment you write the power back out the long way.
By the end of this lesson you will own three rules for working with powers and, just as importantly, you will know which one to reach for. We keep one steady habit of color throughout the whole strand: the letter (the base) is blue, any plain number is amber, and the exponent is purple. When two operations look almost the same, it is the color and the structure — not memorized words — that will keep you safe.
7.4.1 Multiplying powers with the same base
Suppose you multiply a2 by a3. Resist the urge to do anything clever — just write each power out as the repeated multiplication it actually stands for, and count.
a2 · a3
= (a·a)·(a·a·a)
= a·a·a·a·a
= a5
Two copies of a sitting next to three copies of a is simply five copies of a in a row. Nothing was multiplied between the exponents — they were just counted together. The base never changes, because every single factor in the row is the same letter a. So the exponents add:
The copies simply line up end to end. Counting them all is the same as adding the two exponents: 2 + 3 = 5.
Rule 1 — same base, add the exponents
am · an = am+n. When you multiply powers of the same base, keep the base and add the exponents. It works because each power is just a count of how many copies of the base you have, and multiplying puts those copies in one long row.
Worked example
Simplify x4 · x2.
The base is the same letter x in both. Rule 1 applies
Keep the base and add the exponents: 4 + 2 = 6. count the copies
Answer: x6. check: xxxx · xx = six x's
Watch out — don't multiply the exponents here
It is a2 · a3 = a5, nota6. You only multiply exponents in the next rule (a power of a power). Here the powers are side by side, so you add. And the rule needs the same base: a2 · b3 cannot be combined — different letters can't merge into one row.
🎮 Try itSame base: watch the exponents add
Set m and n. The blue tiles are copies of the base a: m of them, then n of them, sliding into one row of m + n.
m2
n3
7.4.2 Raising a power to a power
Now a different question: what happens when a power is itself raised to a power, like (a2)3? The outer exponent 3 is still just a counter, and now it is counting copies of the whole inside thing, a2. So write out three copies of a2 and watch what you get.
(a2)3
= a2·a2·a2
= (aa)(aa)(aa)
= a6
Three groups, each holding two copies of a, makes 3 × 2 = 6 copies in all. This time the exponents multiply — and that is no coincidence. You could finish the middle step with Rule 1 (2 + 2 + 2 = 6), but adding the same number over and over is multiplication. A power of a power is repeated grouping, and repeated grouping multiplies the counts.
Three groups of two copies each is six copies — the exponents multiply: 2 × 3 = 6. Compare this with Rule 1, where the groups were laid in a single row and the exponents added.
Rule 2 — a power of a power, multiply the exponents
(am)n = am·n. When you raise a power to a power, keep the base and multiply the exponents. The outer exponent says "make this many copies of the inside," and each copy brings m factors, so you get m repeated n times.
Worked example
Simplify (y4)2.
A power, y4, is raised to the power 2. Rule 2 applies
Keep the base and multiply the exponents: 4 × 2 = 8. two groups of four copies
Answer: y8. check: y⁴·y⁴ = y⁸ by Rule 1
Watch out — don't add here
(a2)3 = a6, nota5. The brackets are the giveaway: a power inside brackets, with another exponent outside, means multiply. Side-by-side powers (Rule 1) mean add. Same digits, different structure, different answer.
🎮 Try itPower of a power: watch the exponents multiply
Set m and n. You build n dashed groups, each holding m copies of a — for m × n copies in all. Compare with the widget above: same digits, but grouping multiplies.
m (inside)2
n (outside)3
7.4.3 Raising a product to a power
The last rule deals with a product inside the brackets — two (or more) different letters multiplied, then raised to a power, like (ab)3. Once more, the exponent just counts copies, so write three copies of the product ab and then rearrange.
(ab)3
= (ab)(ab)(ab)
= aaa·bbb
= a3b3
Because multiplication can be reordered freely, you may gather all the a's together and all the b's together. Three copies of the product hand you three a's and three b's — so the exponent simply lands on each factor. Think of it as the power being shared out to everyone inside the brackets, the way a doubling recipe doubles every single ingredient.
Reorder the factors and the a's and b's separate cleanly: three of each. The exponent landed on every factor inside the brackets.
Rule 3 — a power of a product, spread it to each factor
(ab)n = anbn. When you raise a product to a power, every factor inside the brackets takes that exponent. It extends to as many factors as you like: (abc)n = anbncn, and it covers numbers too — (2a)3 = 23a3 = 8a3.
Worked example
Simplify (3x)2.
Inside the brackets is a product, 3 times x. Rule 3 applies
Give the exponent 2 to each factor: 32 and x2. don't forget the 3
Work out 32 = 9, so the answer is 9x2. check: (3x)(3x) = 9x²
Watch out — the number gets the exponent too
(3x)2 is 9x2, not3x2. The 3 is a factor inside the brackets, so it must be squared as well: 32 = 9. Leaving the coefficient out is the single most common slip with this rule.
🎮 Try itPower of a product: spread the exponent
Choose how many factors are inside, then set the outer power n. Watch the copies of the product regroup into a run of each factor — every factor ends up with the exponent n.
Factors inside
power n3
7.4.4 Choosing the right rule
You now have all three rules. The real skill — the one this whole lesson is building toward — is deciding which one a problem is asking for, because they can appear in the same expression and they look deceptively alike. Before you touch the exponents, read the structure and ask one question:
▸ Are two powers of the same base multiplied side by side? → ADD the exponents (Rule 1).
▸ Is a power sitting inside brackets with another exponent outside? → MULTIPLY the exponents (Rule 2).
▸ Is a product inside brackets raised to a power? → SPREAD the exponent to each factor (Rule 3).
The two traps are mirror images of each other, and they are worth saying out loud. a2 · a3 is not a6 — multiplied bases add, giving a5. And (a2)3 is not a5 — a power of a power multiplies, giving a6. If you can keep those two straight, you have the chapter.
One glance at the structure decides everything. Multiplied? Add. Power of a power? Multiply. Product raised to a power? Spread it to each factor.
Worked example — a mix in one expression
Simplify x2·x3·(x2)2.
Handle the bracket first with Rule 2: (x2)2 = x4. multiply 2 × 2
The expression is now x2·x3·x4 — three powers multiplied. Rule 1 now
Add the exponents: 2 + 3 + 4 = 9. count all the x's
Answer: x9. do the bracket before the row
Watch out — the two look-alikes
a2·a3 = a5 (add), but (a2)3 = a6 (multiply). The brackets are the difference. When in doubt, write one of them out the long way and count — it takes ten seconds and never lies.
🎮 Try itRule chooser
An expression appears. Decide what to do with the exponents — add, multiply, or spread — then see the worked-out result and the reason. Press New expression for another.
★ Prove it with numbers
A rule about letters can feel like a magic trick until you check it on numbers you can actually compute. Pick a base — say 2 — and the rules become arithmetic you can verify by hand. They had better agree, because letters are only standing in for numbers.
Pick a rule, choose small exponents, and watch both sides get computed all the way to a number. They match exactly — that is the proof.
Rule
m2
n3
§ The three rules, in one breath
A power is repeated multiplication, and every rule comes straight from that. Multiply powers of the same base and you add the exponents (am·an = am+n), because the copies line up in one row. Raise a power to a power and you multiply the exponents ((am)n = amn), because you make groups of groups. Raise a product to a power and the exponent spreads to every factor ((ab)n = anbn). Read the structure first, then act.
Where this goes next — 7.5
These three rules are the toolkit you needed before tackling multiplying expressions. In the next lesson you will multiply whole terms like 2x2 by 5x3: multiply the numbers, then use Rule 1 to handle the letters. Every product of monomials leans on what you just learned here.
✎ Practice 7.4
Read the structure before you act. Work each one out, then open the answer to check.
Simplify a3 · a4.
Show answer
Same base, multiplied side by side → add: 3 + 4 = 7. So a7.
Simplify (a3)4.
Show answer
A power of a power → multiply: 3 × 4 = 12. So a12. (Notice the same digits as Q1 give a different answer — the brackets matter.)
Simplify (xy)3.
Show answer
A product raised to a power → spread the exponent to each factor: x3y3.
Simplify x2 · x3 · x.
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Three powers of x multiplied; remember x = x1. Add: 2 + 3 + 1 = 6. So x6.
Simplify (2a)3.
Show answer
Spread the exponent to both factors: 23 and a3. Since 23 = 8, the answer is 8a3 — don't forget to cube the 2.
True or false: a2 · a3 = a6. If false, give the correct answer.
Show answer
False. Multiplied bases add exponents, not multiply them: a2 · a3 = a5. (You'd get a6 from (a2)3 instead.)
True or false: (a2)3 = a5. If false, give the correct answer.
Show answer
False. A power of a power multiplies exponents: (a2)3 = a6. Writing it out, a2·a2·a2 = six a's.
Simplify y4 · (y2)3.
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Do the bracket first (Rule 2): (y2)3 = y6. Now y4 · y6 (Rule 1): 4 + 6 = 10. So y10.
Simplify (ab2)3.
Show answer
Spread the outer 3 to each factor (Rule 3): a → a3, and b2 → (b2)3 = b6 (Rule 2). So a3b6.
Use base 3 to check that a2·a2 = a4. Compute both sides.
Show answer
Left: 32·32 = 9·9 = 81. Right: 34 = 81. They match, so the rule holds. (And 2 + 2 = 4. ✓)
Simplify (2x2)3.
Show answer
Spread the 3 to each factor: 23 = 8, and (x2)3 = x6. So 8x6.