Exponents

2

Rewrite 9 as 3^2 and 27 as 3^3. Then 3^2 × 3^x = (3^3)^(2x−3), so 3^(x+2) = 3^(6x−9). Equate exponents: x + 2 = 6x − 9, so 11 = 5x and x = 2.

x^0 = 1

Step 1: Apply power law (a^m)^n = a^(mn) to each factor: x^((a-b)(a+b)) × x^((b-c)(b+c)) × x^((c-a)(c+a)) Step 2: Use difference of squares: (a-b)(a+b) = a²-b², (b-c)(b+c) = b²-c², (c-a)(c+a) = c²-a² Step 3: Combine using product law: x^(a²-b² + b²-c² + c²-a²) Step 4: Simplify exponent: a²-b²+b²-c²+c²-a² = 0 Result: x^0 = 1. This is a beautiful algebraic identity involving exponents!

64/729

Step 1: Apply power law: {(-3/2)^(-3)}^2 = (-3/2)^(-3×2) = (-3/2)^(-6) Step 2: Use negative exponent: (-3/2)^(-6) = (-2/3)^6 Step 3: Since the exponent 6 is even, (-2/3)^6 = (2/3)^6 Step 4: (2/3)^6 = 2^6/3^6 = 64/729 Common mistake: Students may forget that a negative base raised to an even power becomes positive. Also, students sometimes apply the outer power incorrectly as multiplying only the inner exponent without flipping the base.

1/9

Step 1: We are given x = 3^m and y = 3^(m+2). Step 2: x/y = 3^m / 3^(m+2) Step 3: Apply quotient law: 3^(m - (m+2)) = 3^(m - m - 2) = 3^(-2) Step 4: 3^(-2) = 1/3² = 1/9 Key insight: The variable m cancels out completely, meaning x/y is always 1/9 regardless of the value of m. Students who substitute a specific value for m often get confused — the algebraic approach is cleaner.