a) i) Evaluate 27^(2/3) ii) Simplify 6(x4)y5 / 12y4y5 iii) Solve p⁻⁸ x p4 = 1/9 iv) Express 6004.83 in…

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a) i) Evaluate 27^(2/3) ii) Simplify 6(x4)y5 / 12y4y5 iii) Solve p⁻⁸ x p4 = 1/9 iv) Express 6004.83 in standard form.

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Haya hapa masuluhisho ya maswali yako: a) i) Evaluate 27^(2/3) Step 1: Andika 27 kama kielelezo cha 3. 27 = 3³; Step 2: Badilisha 27 na 3³ katika usemi na tumia sheria ya vipeo (a^m)ⁿ = a^(mn). 27^(2/3) = (3³)^(2/3) = 3^(3 × 2/3) Step 3: Rahisisha kielelezo na hesabu. 3^(3 × 2/3) = 3² = 9 Jibu ni 9. ii) Simplify (6(y⁴)⁵)/(12y⁴y⁵) Step 1: Rahisisha kielelezo cha juu kwa kutumia sheria (a^m)ⁿ = a^(mn). (6(y⁴)⁵)/(12y⁴y⁵) = (6y^(4 × 5))/(12y⁴y⁵) = (6y²⁰)/(12y⁴y⁵); Step 2: Rahisisha vipeo vya y kwenye denominator kwa kutumia sheria a^m × aⁿ = a^(m+n). (6y²⁰)/(12y⁴y⁵) = (6y²⁰)/(12y⁴⁺⁵) = (6y²⁰)/(12y⁹); Step 3: Rahisisha sehemu ya namba na vipeo vya y kwa kutumia sheria (a^m)/(aⁿ) = a^(m−n). 6/12 × (y²⁰)/(y⁹) = 1/2 × y²⁰⁻⁹ = 1/2y¹¹ Jibu ni 1/2y¹¹. iii) Solve p⁻⁸ × p^(1/4) = 1/9; Step 1: Tumia sheria ya vipeo a^m × aⁿ = a^(m+n) upande wa kushoto. p⁻⁸ × p^(1/4) = p^(−8 + 1/4) Step 2: Rahisisha kielelezo. −8 + 1/4 = −32/4 + 1/4 = −31/4 Hivyo, mlinganyo unakuwa: p^(−31/4) = 1/9 Step 3: Andika 1/9 kama kielelezo cha 3. 1/9 = (1)/(3²) = 3⁻² Hivyo, mlinganyo unakuwa: p^(−31/4) = 3⁻² Step 4: Ili kupata p, inua pande zote mbili kwa kielelezo cha kinyume cha −31/4, ambacho ni −4/31. (p^(−31/4))^(−4/31) = (3⁻²)^(−4/31); p^( (−31/4) × (−4/31) ) = 3^( (−2) × (−4/31) ); p¹ = 3^(8/31); Jibu ni p = 3^(8/31). iv) Express 6004.83 in standard form. Step 1: Hamisha desimali ili namba iwe kati ya 1 na 10. Namba 6004.83 inakuwa 6.00483. Step 2: Hesabu idadi ya sehemu ambazo desimali imesogezwa. Desimali imesogezwa sehemu 3 kwenda kushoto. Hii inamaanisha tunazidisha na 10³. Step 3: Andika namba katika mfumo wa kawaida a × 10ⁿ. 6004.83 = 6.00483 × 10³ Jibu ni 6.00483 × 10³.

A) i) Evaluate 27^(2/3) ii) Simplify 6(x4)y5 / 12y4y5 iii) Solve p⁻⁸ x p4 = 1/9 iv) Express 6004.83 in standard form.

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Step-by-step solution

Part a· 3 steps

Andika 27 kama kielelezo cha 3.

Badilisha 27 na $3^3$ katika usemi na tumia sheria ya vipeo $(a^m)^n = a^{mn}$ .

Rahisisha kielelezo na hesabu.

Part a(ii)· 3 steps
$\frac{1}{2}y^{11}$

Rahisisha kielelezo cha juu kwa kutumia sheria $(a^m)^n = a^{mn}$ .

Rahisisha vipeo vya $y$ kwenye denominator kwa kutumia sheria $a^m \times a^n = a^{m+n}$ .

Rahisisha sehemu ya namba na vipeo vya $y$ kwa kutumia sheria $\frac{a^m}{a^n} = a^{m-n}$ .

Part a(iii)· 4 steps
$p = 3^{\frac{8}{31}}$

Tumia sheria ya vipeo $a^m \times a^n = a^{m+n}$ upande wa kushoto.

Rahisisha kielelezo.

Andika $\frac{1}{9}$ kama kielelezo cha 3.

Ili kupata $p$ , inua pande zote mbili kwa kielelezo cha kinyume cha $-\frac{31}{4}$ , ambacho ni $-\frac{4}{31}$ .

Part a(iv)· 3 steps
$6.00483 \times 10^3$

Hamisha desimali ili namba iwe kati ya 1 na 10.

Hesabu idadi ya sehemu ambazo desimali imesogezwa.

Andika namba katika mfumo wa kawaida $a \times 10^n$ .

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A) i) Evaluate 27^(2/3) ii) Simplify 6(x4)y5 / 12y4y5 iii) Solve p⁻⁸ x p4 = 1/9 iv) Express 6004.83 in standard form.

Answers

Step-by-step solution

Part a· 3 steps

Andika 27 kama kielelezo cha 3.

Badilisha 27 na 333^333 katika usemi na tumia sheria ya vipeo (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Rahisisha kielelezo na hesabu.

Part a(ii)· 3 steps12y11\frac{1}{2}y^{11}21​y11

Rahisisha kielelezo cha juu kwa kutumia sheria (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Rahisisha vipeo vya yyy kwenye denominator kwa kutumia sheria am×an=am+na^m \times a^n = a^{m+n}am×an=am+n.

Rahisisha sehemu ya namba na vipeo vya yyy kwa kutumia sheria aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Part a(iii)· 4 stepsp=3831p = 3^{\frac{8}{31}}p=3318​

Tumia sheria ya vipeo am×an=am+na^m \times a^n = a^{m+n}am×an=am+n upande wa kushoto.