8.0 Review
Laws of Indices
1. Multiplication of same base
1.1 x^m * x^n = x^(m+n)
1.2 Examples:
1.2.1 x^2 * x^3 = x^(2+3) = x^5
1.2.2 y^5 * y^2 = y^(5+2) = y^7
1.2.3 a^4 * a^7 = a^(4+7) = a^11
2. Division of same base
2.1 x^m / x^n = x^(m-n)
2.2 Examples:
2.2.1 x^5 / x^3 = x^(5-3) = x^2
2.2.2 x^7 / x^4 = x^(7-4) = x^3
2.2.3 25 / 2^2 = 25 / 4
3. Power of a power
3.1 (x^m)^n = x^(mn)
3.2 Examples:
3.2.1 (x^3)^2 = x^(32) = x^6
3.2.2 (2^2)^3 = 2^(23) = 64
3.2.3 (5^2)^2 = 5^(22) = 625
4. Fractional powers
4.1 x^(m/n) = n-th root of (x^m) = (n-th root of x)^m
5. Negative powers
5.1 x^(-n) = 1 / x^n
8.1 Problems Related to Indices
Understanding differences
10^3 vs 101010 → same, just expanded form
5^3 = 125, 5^(-3) = 1 / 125 → negative powers give reciprocal
3^6 = 729, 6^3 = 216 → order matters
Key Formulas
x^(-m) = 1 / x^m
x^m * x^n = x^(m+n)
x^m / x^n = x^(m-n)
(x^m)^n = x^(m*n)
Some Important Questions
Example 1: Find the value
4^(-2) = 1 / 16
4^(2/3) = cube root of 16
3^6 = 729
Example 2: Simplify
(x^(a+c) / x^(a+b)) * (x^(b+c) / x^(b+c)) * (x^(c+a) / x^(c+b))
or, apply rules:
x^m / x^n = x^(m-n)
(x^m)^n = x^(mn)
(a+b)(a-b) = a^2 - b^2
or, simplify step-by-step → x^0 = 1
1. Solve 3^x = 81
or, 3^x = 3^4
or, x = 4
2. Solve 8^x = 2^(2x+1)
or, 8^x = (2^3)^x
or, 2^(3x) = 2^(2x+1)
or, 3x = 2x + 1
or, x = 1
3. Solve 3 * 81^x = 9^(x+2)
or, 3 * (3^4)^x = (3^2)^(x+2)
or, 3 * 3^(4x) = 3^(2x+4)
or, 3^(4x+1) = 3^(2x+4)
or, 4x + 1 = 2x + 4
or, 2x = 3
or, x = 1.5
4. Solve 5 * 125^x = 5^(2x-2)
or, 5 * (5^3)^x = 5^(2x-2)
or, 5^(3x+1) = 5^(2x-2)
or, 3x + 1 = 2x - 2
or, x = -3
5. Solve 3^(x+1) + 3^x = 108
or, 3^x * 3 + 3^x = 108
or, 3^x * (3 + 1) = 108
or, 3^x * 4 = 108
or, 3^x = 27
or, x = 3
6. Solve 5^(2x+1) + 5^x = 150
or, 5^(2x+1) = 5 * (5^x)^2
or, 5 * (5^x)^2 + 5^x = 150
or, 5^x * (5 * 5^x + 1) = 150
or, 5^x * 6 = 150
or, 5^x = 25
or, x = 2
7. Solve 3^(x+1) * 2^(2x+1) = 6
or, 3^x * 3 * 2^(2x) * 2 = 6
or, 3^x * 2^(2x) * 6 = 6
or, 3^x * 2^(2x) = 1
or, x = 0
8. Solve 2^(x+3) + 2^x = 18
or, 2^x * 2^3 + 2^x = 18
or, 2^x * (8 + 1) = 18
or, 2^x * 9 = 18
or, 2^x = 2
or, x = 1
9. Solve 2^(2x) + 2^(3x)
or, factor 2^(2x) → 2^(2x) * (1 + 2^x)
or, depends on given value, simplify further using powers
10. Solve 5^(1/x) + 5^(1/x)
or, combine like terms → 2 * 5^(1/x)
or, simplify as needed
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