The three rules for working with powers — the toolkit you need before multiplying expressions.
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.
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:
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.
Simplify x4 · x2.
It is a2 · a3 = a5, not a6. 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.
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.
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.
(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.
Simplify (y4)2.
(a2)3 = a6, not a5. 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.
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.
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.
(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.
Simplify (3x)2.
(3x)2 is 9x2, not 3x2. 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.
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.
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.
Simplify x2·x3·(x2)2.
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.
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.
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.
22 · 23 = 4 · 8 = 32 = 25 ✓ | (23)2 = 82 = 64 = 26 ✓
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.
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.
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.
Read the structure before you act. Work each one out, then open the answer to check.
Powers don't live alone. Before this lesson came 7.3 Adding and Subtracting Expressions: Combining and Clearing Brackets, where you learned to tidy expressions by combining like terms. With the three power rules now in hand, you're ready for 7.5 Multiplying Expressions — where multiplying the numbers and using Rule 1 on the letters lets you multiply whole terms with confidence.