Indices, Surds and Standard Form

Index laws extended to fractional powers, which turns every root into a power — and the surd arithmetic that keeps answers exact instead of rounded.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.2 บทที่ 2 จำนวนจริง (รากที่สอง) และต่อไปยัง ม.4 เลขยกกำลังและกรณฑ์
Common Core (US): 8.EE, N-RN
Cambridge: Lower Secondary Stage 9 — Number: integers, powers and roots

1.1 Fractional indices

Roots written as powers

a^{1/ n} = n a a^{m / n} = n a^{m} = (n a)^{m}

The denominator of the index is the root; the numerator is the power.

ทำไมถึงเป็นแบบนั้น

This is not a new rule bolted on. If $(a^{1/2})^{2} = a^{1} = a$ by the existing index law, then $a^{1/2}$ must be the number whose square is $a$ — which is the square root. The definition is forced by what already holds.

ตัวอย่าง 1

Work out $8^{2/3}$

วิธีทำ

1.Denominator $3$ means cube root; numerator $2$ means square it
2.Take the root first: $38 = 2$
3.Then square: $2^{2}$

ตอบ $4$

ตัวอย่าง 2

Work out $1 6^{- 1/2}$

วิธีทำ

1.The negative index means take the reciprocal
2. $1 6^{1/2} = 4$ , so the answer is $\frac{1}{4}$

ตอบ $\frac{1}{4}$

เคล็ดลับ

Always take the root before applying the power. $8^{2/3}$ done the other way needs $364$ — the same answer from bigger numbers, for no reason.

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