Expressions and Equations

• Exponential equations take the form of y = a * b^x
• Quadratic equations are expressed as y = ax^2

Simplifying Expressions

• 15x^3 / 30x simplifies to x^3 / 2
• 4b^2 / 12b simplifies to b / 3
• 20n^3k / 36nk simplifies to 5n^2 / 9
• (4m^3)^3 simplifies to 64m^9
• n^-3 / n^-7 simplifies to n^4
• 18p^2 / 60p simplifies to 3p / 10
• abc^6 / 23^-5 simplifies to (23^5)abc^6
• 81x^4y^2z^-1 / 54x^3y simplifies to (3x y) / (2z)
• (12pqrs)^2 simplifies to 144p^2q^2r^2s^2
• 32pqrt simplifies as is

Converting Exponential to Radical Form

• 2m^4 * (4m^2)^2 converts to 2m^4 * 16m^4
• n^(1/3) converts to the cube root of n
• n^(1/4) converts to the fourth root of n
• 63 * 10m remains as 63 * 10m
• (6n)^(1/3) converts to the cube root of 6n
• 5^(3/4) converts to the fourth root of 5 cubed
• (10v)^(1/3) converts to the cube root of 10v
• 7k^(1/5) converts to 7 times the fifth root of k

Simplifying with Prime Factorization

• The cube root of 125 simplifies to 5
• The square root of 49 simplifies to 7
• The square root of 98 simplifies to 7√2
• The square root of 320 simplifies to 8√5
• 4000005 remains as 4000005
• 325 remains as 325
• The cube root of 343 simplifies to 7

Simplifying Radicals with Prime Factorization

• √108 simplifies to 6√3
• √112 simplifies to 4√7
• √36 simplifies to 6
• √48 simplifies to 4√3
• √384 simplifies to 8√6
• √27 simplifies to 3√3
• √175 simplifies to 5√7

Scientific Notation

• 4.34 × 10^3 = 4340
• 3.66 × 10^-5 = 0.0000366
• 2.3 × 10^4 = 23000
• 4.04 × 10^-1 = 0.404
• (9.0001 × 10^7) * (1.3 × 10^7) = 1.170013 × 10^15
• (1.011 × 10^-16) * (5.4321 × 10^-2) = 5.4918531 × 10^-18

Exponential to Logarithmic Form

• 2^1 = 25 expressed as log base 2 of 25 = 1
• y^-4 = x expressed as log base y of x = -4
• (1/8)^(2y) = (x+1) expressed as log base 1/8 of (x+1) = 2y
• a^-3 = (bx)/y expressed as log base a of (bx)/y = -3

Logarithmic to Exponential Form

• log base 4 of 64 = 3 expressed as 4^3 = 64
• log base (m+1) of (n-1) = 3mn expressed as (m+1)^(3mn) = n-1
• log base 16 of y = (12k)^x expressed as 16^((12k)^x) = y

Solving Equations

• 32 - 2n = 3 is solved for n
• 23k = 8 is solved for k
• 10x = 19 is solved for x
• 20b = 14 is solved for b
• 20k - 6 + 7 = 48 is solved for k
• 14 - p - 9 = 33 is solved for p
• 94n + 8 = 32 is solved for n
• -10 * 7n - 2 = -50 is solved for n
• 5 * 2x + 5 = 100 is solved for x
• -9 + ln(-10r) = -10 is solved for r
• 7ln(-2x) = 28 is solved for x

Solving Logarithmic Equations

• log(m) = 2 is solved for m
• log(x+6) - 1 = 2 is solved for x
• log8x - 10 = -12 is solved for x
• log base x of 25 = 2 is solved for x
• log base 8 of r = -1 is solved for r
• -7log base 2 of v = 7 is solved for v

Rewriting as a Single Logarithm

• log base 8 of 6 + log base 8 of 5 = log base 8 of (6*5)
• 3log base 8 of 10 + 2log base 8 of 7 = log base 8 of (10^3 * 7^2)
• log base 5 of a + log base 5 of b + 6log base 5 of c = log base 5 of (a * b * c^6)
• 6log base 5 of x - 18log base 5 of y = log base 5 of (x^6 / y^18)

Approximating Logarithms

Given Approximations:

• log base 3 of 10 ≈ 2.1
• log base 3 of 8 ≈ 1.9
• log base 3 of 7 ≈ 1.8
• log base 4 of 10 ≈ 1.7
• log base 4 of 6 ≈ 1.3
• log base 4 of 7 ≈ 1.4
• log base 7 of 10 ≈ 1.2
• log base 7 of 3 ≈ 0.6
• log base 7 of 8 ≈ 1.1
• log base 9 of 11 ≈ 1.1
• log base 9 of 6 ≈ 0.8
• log base 9 of 8 ≈ 0.9

Simplify expressions using rules of exponents. Exponential equations take the form of y = a * b^x and Quadratic equations are expressed as y = ax^2. Convert exponential to radical form.