Indices, Surds and Polynomials

Fractional and negative indices without a calculator, surds left in exact form, and the factor and remainder theorems — the tools that let you factorise a cubic.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.4 เลขยกกำลังและกรณฑ์ (เพิ่มเติม) และเรื่องทฤษฎีบทเศษเหลือในพหุนาม
Common Core (US): N-RN, A-APR, A-SSE
Cambridge: IGCSE 0606 — Topics 3 and 4: Factors of polynomials; Equations, inequalities and graphs

3.1 Indices and surds in exact form

The index laws, including the two 0580 only touched

a^{- n} = \frac{1}{a ^{n}} a^{1/ n} = n a a^{m / n} = (n a)^{m} a^{0} = 1

Taking the root FIRST keeps the numbers small: $8^{2/3}$ is $2^{2} = 4$ , not the cube root of $64$ found the hard way.

ตัวอย่าง 1

Evaluate $(\frac{16}{81})^{- 3/4}$ without a calculator.

วิธีทำ

1.Negative index: turn the fraction over — $(\frac{81}{16})^{3/4}$
2.Fourth roots: $\frac{3}{2}$
3.Cube it: $\frac{27}{8}$

ตอบ $\frac{27}{8}$

ตัวอย่าง 2

Simplify $\frac{4}{3 - 5}$ , giving your answer in the form $a + b 5$ .

วิธีทำ

1.Multiply top and bottom by the conjugate $3 + 5$
2.Denominator: $9 - 5 = 4$
3.Numerator: $12 + 45$

ตอบ $3 + 5$

ข้อควรระวัง

$a + b$ is not $a + b$ . Try $a = 9$ and $b = 16$ : the left side is $5$ and the right side is $7$ . The root sign distributes over multiplication only, and no amount of familiarity makes the other version true.

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