Indices, Roots and Standard Form

The laws that let you handle enormous and tiny numbers in one line, and the notation astronomers and chemists use for the same reason.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.2 บทที่ 5 สมบัติของเลขยกกำลัง
Common Core (US): 8.EE
Cambridge: Lower Secondary Stage 8 — Number: integers, powers and roots

1.1 The laws of indices

The three that do the work

a^{m} \times a^{n} = a^{m + n} \frac{a ^{m}}{a ^{n}} = a^{m - n} (a^{m})^{n} = a^{mn}

All three need the SAME base. There is no law for adding powers of different bases, because there is nothing to combine.

ทำไมถึงเป็นแบบนั้น

None of this is worth memorising if you can write it out. $a^{3} \times a^{2} = (aaa) (aa) = a^{5}$ : the factors are being collected, so the indices add. Division cancels them, so the indices subtract.

ตัวอย่าง 1

Simplify $\frac{5 ^{7} \times 5 ^{2}}{5 ^{4}}$

วิธีทำ

1.Numerator: $5^{7 + 2} = 5^{9}$
2.Then divide: $5^{9 - 4}$

ตอบ $5^{5} = 3, 125$

ตัวอย่าง 2

Simplify $(2 x^{3})^{4}$

วิธีทำ

1.The power applies to both parts inside the bracket
2. $2^{4} = 16$ and $(x^{3})^{4} = x^{12}$

ตอบ $16 x^{12}$

ข้อควรระวัง

$2^{3} \times 3^{2}$ cannot be combined — the bases differ, so work each out and multiply: $8 \times 9 = 72$ . Reaching for an index law here is the most common slip in this chapter.

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