Question 1. a Simplify (81/256)⁻³/4 b x^m+2n. x3m−6n c If n = 3^(2/3) + 3^(−2/3), show that 9n2 - 27n = 82

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let's tackle these exponent and algebra problems. a) To simplify (81/256)^(−3/4): Step 1: Apply the negative exponent rule, which states a⁻ⁿ = (1)/(aⁿ) or (a/b)⁻ⁿ = (b/a)ⁿ. (81/256)^(−3/4) = (256/81)^(3/4); Step 2: Express the base numbers as powers. 256 = 4⁴ and 81 = 3⁴. (256/81)^(3/4) = ((4⁴)/(3⁴))^(3/4) = ((4/3)⁴)^(3/4); Step 3: Apply the power of a power rule, (a^m)ⁿ = a^(mn). ((4/3)⁴)^(3/4) = (4/3)^(4 · 3/4) = (4/3)³ Step 4: Calculate the final power. (4/3)³ = (4³)/(3³) = 64/27 The simplified expression is 64/27. b) To simplify (x^(m+2n) · x^(3m−8n))/(x^(5m−6n)): Step 1: Apply the product rule for exponents in the numerator, a^p · a^q = a^(p+q). x^(m+2n) · x^(3m−8n) = x^((m+2n) + (3m−8n)) = x^(m+3m+2n−8n) = x^(4m−6n); Step 2: Apply the quotient rule for exponents, (a^p)/(a^q) = a^(p−q). (x^(4m−6n))/(x^(5m−6n)) = x^((4m−6n) − (5m−6n)) Step 3: Simplify the exponent. x^(4m−6n − 5m + 6n) = x^(4m−5m − 6n+6n) = x^(−m) Step 4: Express with a positive exponent. x^(−m) = (1)/(x^m) The simplified expression is (1)/(x^m). c) If n = 3^(2/3) + 3^(−2/3), show that 9n² − 27n = 82. Note: There appears to be a typo in the question. Based on the common structure of such problems and the derivation below, the equation to be shown is likely 9n³ − 27n = 82, not 9n² − 27n = 82. Also, the variable in the equation should be n, not x. I will proceed with showing 9n³ − 27n = 82 assuming these corrections. Step 1: Start with the given expression for n. n = 3^(2/3) + 3^(−2/3); Step 2: Cube both sides of the equation. Recall the identity (a+b)³ = a³ + b³ + 3ab(a+b). Let a = 3^(2/3) and b = 3^(−2/3). n³ = (3^(2/3))³ + (3^(−2/3))³ + 3(3^(2/3))(3^(−2/3))(3^(2/3) + 3^(−2/3)) Step 3: Simplify each term. (3^(2/3))³ = 3^((2/3) · 3) = 3² = 9; (3^(−2/3))³ = 3^((−2/3) · 3) = 3⁻² = 1/9; 3(3^(2/3))(3^(−2/3)) = 3 · 3^((2/3 − 2/3)) = 3 · 3⁰ = 3 · 1 = 3 The term (3^(2/3) + 3^(−2/3)) is equal to n. Step 4: Substitute the simplified terms back into the equation for n³. n³ = 9 + 1/9 + 3n Step 5: Rearrange the equation to isolate the terms involving n. n³ − 3n = 9 + 1/9; n³ − 3n = 81/9 + 1/9; n³ − 3n = 82/9 Step 6: Multiply the entire equation by 9 to clear the denominator. 9(n³ − 3n) = 9 (82/9); 9n³ − 27n = 82; Proven (assuming typo corrected to 9n³ − 27n = 82)

Question 1. a Simplify (81/256)⁻³/4 b x^m+2n. x3m-6n c If n = 3^(2/3) + 3^(-2/3), show that 9n2 − 27n = 82

Answers

Step-by-step solution

Part a· 4 steps

Apply the negative exponent rule, which states $a^{-n} = \frac{1}{a^n}$ or $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$ .

Express the base numbers as powers.

Apply the power of a power rule, $(a^m)^n = a^{mn}$ .

Calculate the final power.

Part b· 4 steps
$\frac{1}{x^m}$

Apply the product rule for exponents in the numerator, $a^p \cdot a^q = a^{p+q}$ .

Apply the quotient rule for exponents, $\frac{a^p}{a^q} = a^{p-q}$ .

Simplify the exponent.

Express with a positive exponent.

Part c· 6 steps
$\text{Proven (assuming typo corrected to } 9n^3 - 27n = 82)$

Start with the given expression for $n$ .

Cube both sides of the equation.

Simplify each term.

Substitute the simplified terms back into the equation for $n^3$ .

Rearrange the equation to isolate the terms involving $n$ .

Multiply the entire equation by 9 to clear the denominator.

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